Pendulum (mechanics)

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Animation of a pendulum showing the velocity and acceleration vectors.

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

Simple gravity pendulum

A simple gravity pendulum is an idealized mathematical model of a real pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. The model is based on these assumptions:

• The rod or cord on which the bob swings is massless, inextensible and always remains taut.
• The bob is a point mass.
• Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
• The motion does not lose energy to friction or air resistance.
• The gravitational field is uniform.
• The support does not move.

The differential equation which represents the motion of a simple pendulum is

$${\displaystyle \frac{d^2\theta }{d{t}^2}+\frac{g}{\ell }\sin \theta =0}$$

(Eq. 1)

where g is the magnitude of the gravitational field, ℓ is the length of the rod or cord, and θ is the angle from the vertical to the pendulum.

"Force" derivation of (Eq. 1)

Figure 1. Force diagram of a simple gravity pendulum.

Consider Figure 1 on the right, which shows the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle θ is measured in radians, and this is crucial for this formula. The blue arrow is the gravitational force acting on the bob, and the violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion. The direction of the bob's instantaneous velocity always points along the red axis, which is considered the tangential axis because its direction is always tangent to the circle. Consider Newton's second law,

$${\displaystyle F=ma}$$

where F is the sum of forces on the object, m is mass, and a is the acceleration. Because we are only concerned with changes in speed, and because the bob is forced to stay in a circular path, we apply Newton's equation to the tangential axis only. The short violet arrow represents the component of the gravitational force in the tangential axis, and trigonometry can be used to determine its magnitude. Thus,

$${\displaystyle \begin{array}{rl}F& =-mg\sin \theta =ma,\text{so}\\ a& =-g\sin \theta ,\end{array}}$$

where g is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that θ and a always point in opposite directions. This makes sense because when a pendulum swings further to the left, we would expect it to accelerate back toward the right.

This linear acceleration a along the red axis can be related to the change in angle θ by the arc length formulas; s is arc length:

$${\displaystyle \begin{array}{rl}s& =\ell \theta ,\\ v& =\frac{ds}{dt}=\ell \frac{d\theta }{dt},\\ a& =\frac{d^{2}s}{d{t}^2}=\ell \frac{d^2\theta }{d{t}^2},\end{array}}$$

thus:

$${\displaystyle \begin{array}{rl}\ell \frac{d^2\theta }{d{t}^2}& =-g\sin \theta ,\\ \frac{d^2\theta }{d{t}^2}+\frac{g}{\ell }\sin \theta & =0.\end{array}}$$

"Torque" derivation of (Eq. 1)

Equation (1) can be obtained using two definitions for torque.

$${\displaystyle \mathit{\tau }=\mathbf{r}\times \mathbf{F}=\frac{d\mathbf{L}}{dt}.}$$

First start by defining the torque on the pendulum bob using the force due to gravity.

$${\displaystyle \mathit{\tau }=\mathbf{l}\times {\mathbf{F}}_{\mathrm{g}},}$$

where l is the length vector of the pendulum and F_g is the force due to gravity.

For now just consider the magnitude of the torque on the pendulum.

$${\displaystyle |\mathit{\tau }|=-mg\ell \sin \theta ,}$$

where m is the mass of the pendulum, g is the acceleration due to gravity, l is the length of the pendulum, and θ is the angle between the length vector and the force due to gravity.

Next rewrite the angular momentum.

$${\displaystyle \mathbf{L}=\mathbf{r}\times \mathbf{p}=m\mathbf{r}\times (\mathit{\omega }\times \mathbf{r}).}$$

Again just consider the magnitude of the angular momentum.

$${\displaystyle |\mathbf{L}|=m{r}^2\omega =m{\ell }^2\frac{d\theta }{dt}.}$$

and its time derivative

$${\displaystyle \frac{d}{dt}|\mathbf{L}|=m{\ell }^2\frac{d^2\theta }{d{t}^2},}$$

According to τ = dL/dt, we can get by comparing the magnitudes

$${\displaystyle -mg\ell \sin \theta =m{\ell }^2\frac{d^2\theta }{d{t}^2},}$$

thus:

$${\displaystyle \frac{d^2\theta }{d{t}^2}+\frac{g}{\ell }\sin \theta =0,}$$

which is the same result as obtained through force analysis.

"Energy" derivation of (Eq. 1)

Figure 2. Trigonometry of a simple gravity pendulum.

It can also be obtained via the conservation of mechanical energy principle: any object falling a vertical distance ${\displaystyle h}$ would acquire kinetic energy equal to that which it lost to the fall. In other words, gravitational potential energy is converted into kinetic energy. Change in potential energy is given by

$${\displaystyle \mathrm{\Delta }U=mgh.}$$

The change in kinetic energy (body started from rest) is given by

$${\displaystyle \mathrm{\Delta }K=\frac{1}{2}m{v}^2.}$$

Since no energy is lost, the gain in one must be equal to the loss in the other

$${\displaystyle \frac{1}{2}m{v}^2=mgh.}$$

The change in velocity for a given change in height can be expressed as

$${\displaystyle v=\sqrt{2gh}.}$$

Using the arc length formula above, this equation can be rewritten in terms of dθ/dt:

$${\displaystyle \begin{array}{rl}v=\ell \frac{d\theta }{dt}& =\sqrt{2gh},\text{so}\\ \frac{d\theta }{dt}& =\frac{\sqrt{2gh}}{\ell },\end{array}}$$

where h is the vertical distance the pendulum fell. Look at Figure 2, which presents the trigonometry of a simple pendulum. If the pendulum starts its swing from some initial angle θ₀, then y₀, the vertical distance from the screw, is given by

$${\displaystyle y_0=\ell \cos {\theta }_0.}$$

Similarly, for y₁, we have

$${\displaystyle y_1=\ell \cos \theta .}$$

Then h is the difference of the two

$${\displaystyle h=\ell \left(\cos \theta -\cos {\theta }_0\right).}$$

In terms of dθ/dt gives

$${\displaystyle \frac{d\theta }{dt}=\sqrt{\frac{2g}{\ell }(\cos \theta -\cos {\theta }_0)}.}$$

(Eq. 2)

This equation is known as the first integral of motion, it gives the velocity in terms of the location and includes an integration constant related to the initial displacement (θ₀). We can differentiate, by applying the chain rule, with respect to time to get the acceleration

$${\displaystyle \begin{array}{rl}\frac{d}{dt}\frac{d\theta }{dt}& =\frac{d}{dt}\sqrt{\frac{2g}{\ell }\left(\cos \theta -\cos {\theta }_0\right)},\\ \frac{d^2\theta }{d{t}^2}& =\frac{1}{2}\frac{-\frac{2g}{\ell }\sin \theta }{\sqrt{\frac{2g}{\ell }(\cos \theta -\cos {\theta }_0)}}\frac{d\theta }{dt}\\ & =\frac{1}{2}\frac{-\frac{2g}{\ell }\sin \theta }{\sqrt{\frac{2g}{\ell }(\cos \theta -\cos {\theta }_0)}}\sqrt{\frac{2g}{\ell }(\cos \theta -\cos {\theta }_0)}=-\frac{g}{\ell }\sin \theta ,\\ \frac{d^2\theta }{d{t}^2}& +\frac{g}{\ell }\sin \theta =0,\end{array}}$$

which is the same result as obtained through force analysis.

Small-angle approximation

Small-angle approximation for the sine function: For θ ≈ 0 we find sin θ ≈ θ.

The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian (often cited as less than 0.1 radians, about 6°), or

$${\displaystyle \theta \ll 1,}$$

then substituting for sin θ into Eq. 1 using the small-angle approximation,

$${\displaystyle \sin \theta \approx \theta ,}$$

yields the equation for a harmonic oscillator,

$${\displaystyle \frac{d^2\theta }{d{t}^2}+\frac{g}{\ell }\theta =0.}$$

The error due to the approximation is of order θ³ (from the Taylor expansion for sin θ).

Let the starting angle be θ₀. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes

${\displaystyle \theta (t)={\theta }_0\cos \left(\sqrt{\frac{g}{\ell }}t\right){\theta }_0\ll 1.}$

The motion is simple harmonic motion where θ₀ is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate period of the motion is then

${\displaystyle T_0=2\pi \sqrt{\frac{\ell }{g}}{\theta }_0\ll 1}$

which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ₀; this is the property of isochronism that Galileo discovered.

Rule of thumb for pendulum length

$${\displaystyle T_0=2\pi \sqrt{\frac{\ell }{g}}}$$

gives

$${\displaystyle \ell =\frac{g}{{\pi }^2}\frac{T_0^2}{4}.}$$

If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then g ≈ 9.81 m/s², and g/π² ≈ 1 m/s² (0.994 is the approximation to 3 decimal places).

Therefore, relatively reasonable approximations for the length and period are:

$${\displaystyle \begin{array}{rl}\ell & \approx \frac{T_0^2}{4},\\ T_0& \approx 2\sqrt{\ell }\end{array}}$$

where T₀ is the number of seconds between two beats (one beat for each side of the swing), and l is measured in metres.

Arbitrary-amplitude period

Figure 3. Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained numerically evaluating the elliptic integral.

Figure 4. Relative errors using the power series for the period.

Figure 5. Potential energy and phase portrait of a simple pendulum. Note that the x-axis, being angle, wraps onto itself after every 2π radians.

For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (Eq. 2),

$${\displaystyle \frac{dt}{d\theta }=\sqrt{\frac{\ell }{2g}}\frac{1}{\sqrt{\cos \theta -\cos {\theta }_0}}}$$

and then integrating over one complete cycle,

$${\displaystyle T=t({\theta }_0\to 0\to -{\theta }_0\to 0\to {\theta }_0),}$$

or twice the half-cycle

$${\displaystyle T=2t({\theta }_0\to 0\to -{\theta }_0),}$$

or four times the quarter-cycle

$${\displaystyle T=4t({\theta }_0\to 0),}$$

which leads to

$${\displaystyle T=4\sqrt{\frac{\ell }{2g}}{\int }_0^{{\theta }_0}\frac{d\theta }{\sqrt{\cos \theta -\cos {\theta }_0}}.}$$

Note that this integral diverges as θ₀ approaches the vertical

$${\displaystyle \underset{{\theta }_0\to \pi }{lim}T=\mathrm{\infty },}$$

so that a pendulum with just the right energy to go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.)

This integral can be rewritten in terms of elliptic integrals as

$${\displaystyle T=4\sqrt{\frac{\ell }{g}}F\left(\frac{\pi }{2},\sin \frac{{\theta }_0}{2}\right)}$$

where F is the incomplete elliptic integral of the first kind defined by

$${\displaystyle F(\phi ,k)={\int }_0^{\phi }\frac{du}{\sqrt{1-k^2{\sin }^{2}u}}.}$$

Or more concisely by the substitution

$${\displaystyle \sin u=\frac{\sin \frac{\theta }{2}}{\sin \frac{{\theta }_0}{2}}}$$

expressing θ in terms of u,

${\displaystyle T=\frac{2T_0}{\pi }K(k),\text{where}k=\sin \frac{{\theta }_0}{2}.}$ Eq. 3

Here K is the complete elliptic integral of the first kind defined by

$${\displaystyle K(k)=F\left(\frac{\pi }{2},k\right)={\int }_0^{\frac{\pi }{2}}\frac{du}{\sqrt{1-k^2{\sin }^{2}u}}.}$$

For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 9.80665 m/s²) at initial angle 10 degrees is

$${\displaystyle 4\sqrt{\frac{1\text{ m}}{g}}K\left(\sin \frac{{10}^{\circ }}{2}\right)\approx 2.0102\text{ s}.}$$

The linear approximation gives

$${\displaystyle 2\pi \sqrt{\frac{1\text{ m}}{g}}\approx 2.0064\text{ s}.}$$

The difference between the two values, less than 0.2%, is much less than that caused by the variation of g with geographical location.

From here there are many ways to proceed to calculate the elliptic integral.

Legendre polynomial solution for the elliptic integral

Given Eq. 3 and the Legendre polynomial solution for the elliptic integral:

$${\displaystyle K(k)=\frac{\pi }{2}\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{(2n-1)!!}{(2n)!!}{k}^n\right)}^2}$$

where n!! denotes the double factorial, an exact solution to the period of a simple pendulum is:

$${\displaystyle \begin{array}{rl}T& =2\pi \sqrt{\frac{\ell }{g}}\left(1+{\left(\frac{1}{2}\right)}^2{\sin }^2\frac{{\theta }_0}{2}+{\left(\frac{1\cdot 3}{2\cdot 4}\right)}^2{\sin }^4\frac{{\theta }_0}{2}+{\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)}^2{\sin }^6\frac{{\theta }_0}{2}+\cdots \right)\\ & =2\pi \sqrt{\frac{\ell }{g}}\cdot \sum _{n=0}^{\mathrm{\infty }}\left({\left(\frac{(2n)!}{(2^n\cdot n!{)}^2}\right)}^2\cdot {\sin }^{2n}\frac{{\theta }_0}{2}\right).\end{array}}$$

Figure 4 shows the relative errors using the power series. T₀ is the linear approximation, and T₂ to T₁₀ include respectively the terms up to the 2nd to the 10th powers.

Power series solution for the elliptic integral

Another formulation of the above solution can be found if the following Maclaurin series:

$${\displaystyle \sin \frac{{\theta }_0}{2}=\frac{1}{2}{\theta }_0-\frac{1}{48}{\theta }_0^3+\frac{1}{3840}{\theta }_0^5-\frac{1}{645120}{\theta }_0^7+\cdots .}$$

is used in the Legendre polynomial solution above. The resulting power series is:

$${\displaystyle T=2\pi \sqrt{\frac{\ell }{g}}\left(1+\frac{1}{16}{\theta }_0^2+\frac{11}{3072}{\theta }_0^4+\frac{173}{737280}{\theta }_0^6+\frac{22931}{1321205760}{\theta }_0^8+\frac{1319183}{951268147200}{\theta }_0^{10}+\frac{233526463}{2009078326886400}{\theta }_0^{12}+\cdots \right),}$$

more fractions available in the On-Line Encyclopedia of Integer Sequences with OEIS: A223067 having the numerators and OEIS: A223068 having the denominators.

Arithmetic-geometric mean solution for elliptic integral

Given Eq. 3 and the arithmetic–geometric mean solution of the elliptic integral:

$${\displaystyle K(k)=\frac{\pi }{2M(1-k,1+k)},}$$

where M(x,y) is the arithmetic-geometric mean of x and y.

This yields an alternative and faster-converging formula for the period:

$${\displaystyle T=\frac{2\pi }{M\left(1,\cos \frac{{\theta }_0}{2}\right)}\sqrt{\frac{\ell }{g}}.}$$

The first iteration of this algorithm gives

$${\displaystyle T_1=\frac{2T_0}{1+\cos \frac{{\theta }_0}{2}}.}$$

This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since $\frac{1}{2}\left(1+\cos \left(\frac{{\theta }_0}{2}\right)\right)={\cos }^2\frac{{\theta }_0}{4},$ the expression can be written more concisely as

$${\displaystyle T_1=T_0{\sec }^2\frac{{\theta }_0}{4}.}$$

The second order expansion of ${\displaystyle {\sec }^2({\theta }_0/4)}$ reduces to $T\approx T_0\left(1+\frac{{\theta }_0^2}{16}\right).$

A second iteration of this algorithm gives

$${\displaystyle T_2=\frac{4T_0}{1+\cos \frac{{\theta }_0}{2}+2\sqrt{\cos \frac{{\theta }_0}{2}}}=\frac{4T_0}{{\left(1+\sqrt{\cos \frac{{\theta }_0}{2}}\right)}^2}.}$$

This second approximation has a relative error of less than 1% for angles up to 163.10 degrees.

Approximate formulae for the nonlinear pendulum period

Though the exact period ${\displaystyle T}$ can be determined, for any finite amplitude ${\displaystyle {\theta }_0<\pi }$ rad, by evaluating the corresponding complete elliptic integral ${\displaystyle K(k)}$, where ${\displaystyle k\equiv \sin ({\theta }_0/2)}$, this is often avoided in applications because it is not possible to express this integral in a closed form in terms of elementary functions. This has made way for research on simple approximate formulae for the increase of the pendulum period with amplitude (useful in introductory physics labs, classical mechanics, electromagnetism, acoustics, electronics, superconductivity, etc. The approximate formulae found by different authors can be classified as follows:

• ‘Not so large-angle’ formulae, i.e. those yielding good estimates for amplitudes below ${\displaystyle \pi /2}$ rad (a natural limit for a bob on the end of a flexible string), though the deviation with respect to the exact period increases monotonically with amplitude, being unsuitable for amplitudes near to ${\displaystyle \pi }$ rad. One of the simplest formulae found in literature is the following one by Lima (2006): $T\approx -T_0\frac{\ln a}{1-a}$, where ${\displaystyle a\equiv \cos ({\theta }_0/2)}$.
• ‘Very large-angle’ formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to ${\displaystyle \pi }$rad, with an error that increases monotonically for smaller amplitudes (i.e., unsuitable for small amplitudes). One of the better such formulae is that by Cromer, namely: $T\approx \frac{2}{\pi }{T}_0\ln (4/a)$.

Of course, the increase of ${\displaystyle T}$ with amplitude is more apparent when ${\displaystyle \pi /2<{\theta }_0<\pi }$, as has been observed in many experiments using either a rigid rod or a disc. As accurate timers and sensors are currently available even in introductory physics labs, the experimental errors found in ‘very large-angle’ experiments are already small enough for a comparison with the exact period and a very good agreement between theory and experiments in which friction is negligible has been found. Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. In 2008, Lima derived a weighted-average formula with this characteristic:

$${\displaystyle T\approx \frac{r{a}^2{T}_{\text{Lima}}+k^2{T}_{\text{Cromer}}}{r{a}^2+k^2},}$$

where ${\displaystyle r=7.17}$, which presents a maximum error of only 0.6% (at ${\displaystyle {\theta }_0={95}^{\circ }}$).

Arbitrary-amplitude angular displacement

The Fourier series expansion of ${\displaystyle \theta (t)}$ is given by

$${\displaystyle \theta (t)=8\sum _{n\ge 1\text{ odd}}\frac{(-1{)}^{\lfloor n/2\rfloor }}{n}\frac{q^{n/2}}{1+q^n}\cos (n\omega t)}$$

where ${\displaystyle q}$ is the elliptic nome, ${\displaystyle q=\exp \left(-\pi K(\sqrt{1-k^2})/K(k)\right),}$ ${\displaystyle k=\sin ({\theta }_0/2),}$ and ${\displaystyle \omega =2\pi /T}$ the angular frequency.

If one defines

$${\displaystyle \epsilon =\frac{1}{2}\cdot \frac{1-\sqrt{\cos ({\theta }_0/2)}}{1+\sqrt{\cos ({\theta }_0/2)}}}$$

${\displaystyle q}$ can be approximated using the expansion

$${\displaystyle q=\epsilon +2{\epsilon }^5+15{\epsilon }^9+150{\epsilon }^{13}+1707{\epsilon }^{17}+20910{\epsilon }^{21}+\cdots }$$

(see OEIS: A002103). Note that for ${\displaystyle {\theta }_0<\pi }$ we have ${\displaystyle \epsilon <\frac{1}{2}}$, thus the approximation is applicable even for large amplitudes.

Equivalently, the angle can be given in terms of the Jacobi elliptic function ${\displaystyle \mathrm{cd}}$ with modulus ${\displaystyle k}$

$${\displaystyle \theta (t)=2\arcsin \left(k\mathrm{cd}\left(\sqrt{\frac{g}{\ell }}t;k\right)\right),k=\sin \frac{{\theta }_0}{2}.}$$

For small ${\displaystyle x}$, ${\displaystyle \sin x\approx x}$, ${\displaystyle \arcsin x\approx x}$ and ${\displaystyle \mathrm{cd}(t;0)=\cos t}$, so the solution is well-approximated by the solution given in Pendulum (mechanics)#Small-angle approximation.

Examples

The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity. With a large enough initial velocity the pendulum does not oscillate back and forth but rotates completely around the pivot.

Damped, driven pendulum

The above discussion focuses on a pendulum bob only acted upon by the force of gravity. Here we instead suppose a damping force, e.g. air resistance, as well as a sinusoidal driving force acts on the body. This system is a damped, driven oscillator, and is chaotic.

Equation (1) can be written as

$${\displaystyle m{l}^2\frac{d^2\theta }{d{t}^2}=-mgl\sin \theta }$$

(see the Torque derivation of Equation (1) above).

We can now add a damping term and a forcing term to the right hand side to get

$${\displaystyle m{l}^2\frac{d^2\theta }{d{t}^2}=-mgl\sin \theta -b\frac{d\theta }{dt}+a\cos (\mathrm{\Omega }t)}$$

where we have assumed the damping couple to be proportional to the angular velocity (this is true for low-speed air resistance, see also Drag (physics)). ${\displaystyle a}$ and ${\displaystyle b}$ are constants defining the amplitude of forcing and the degree of damping respectively. $\mathrm{\Omega }$ is the angular frequency of the driving oscillations.

We can now simplify by dividing through by $m{l}^2$ and redefining the constants appropriately to obtain

$${\displaystyle \frac{d^2\theta }{d{t}^2}+B\frac{d\theta }{dt}+\frac{g}{l}\sin \theta =A\cos (\mathrm{\Omega }t).}$$

This equation exhibits chaotic behaviour. The exact motion of this pendulum can only be found numerically and is highly dependent on initial conditions, e.g. the initial velocity and the starting amplitude. However, the small angle approximation outlined above can still be used under the required conditions to give an approximate analytical solution.

Compound pendulum

A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot ${\displaystyle O}$. In this case the pendulum's period depends on its moment of inertia ${\displaystyle I_O}$ around the pivot point.

The equation of torque gives:

$${\displaystyle \tau =I\alpha }$$

where: ${\displaystyle \alpha }$ is the angular acceleration. ${\displaystyle \tau }$ is the torque

The torque is generated by gravity so: ${\displaystyle \tau =-mg{r}_{\mathrm{C}\mathrm{M}}\sin \theta }$ where:

• ${\displaystyle m}$ is the total mass of the ridig body (rod and bob)
• ${\displaystyle r_{\mathrm{C}\mathrm{M}}}$ is the distance from the pivot point to the system's center of mass
• ${\displaystyle \theta }$ is the angle from the vertical

Hence, under the small-angle approximation, ${\displaystyle \sin \theta \approx \theta }$,

$${\displaystyle \alpha =\ddot{\theta }\approx -\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}\theta }$$

where ${\displaystyle I_O}$ is the moment of inertia of the body about the pivot point ${\displaystyle O}$.

The expression for ${\displaystyle \alpha }$ is of the same form as the conventional simple pendulum and gives a period of^[2]

$${\displaystyle T=2\pi \sqrt{\frac{I_O}{mg{r}_{\mathrm{C}\mathrm{M}}}}}$$

And a frequency of

$${\displaystyle f=\frac{1}{T}=\frac{1}{2\pi }\sqrt{\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}}}$$

If the initial angle is taken into consideration (for large amplitudes), then the expression for ${\displaystyle \alpha }$ becomes:

$${\displaystyle \alpha =\ddot{\theta }=-\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}\sin \theta }$$

and gives a period of:

$${\displaystyle T=4\mathrm{K}\left({\sin }^2\frac{{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\right)\sqrt{\frac{I_O}{mg{r}_{\mathrm{C}\mathrm{M}}}}}$$

where ${\displaystyle {\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}$ is the maximum angle of oscillation (with respect to the vertical) and ${\displaystyle \mathrm{K}(k)}$ is the complete elliptic integral of the first kind.

An important concept is the equivalent length, ${\displaystyle {\ell }^{\mathrm{e}\mathrm{q}}}$, the length of a simple pendulums that has the same angular frequency ${\displaystyle {\omega }_0}$ as the compound pendulum: ${\displaystyle {{\omega }_0}^2=\frac{g}{{\ell }^{\mathrm{e}\mathrm{q}}}=\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}⟹{\ell }^{\mathrm{e}\mathrm{q}}=\frac{I_O}{g{r}_{\mathrm{C}\mathrm{M}}}}$

Consider the following cases:

• The simple pendulum is the special case where all the mass is located at the bob swinging at a distance ${\displaystyle \ell }$ from the pivot. Thus, ${\displaystyle r_{\mathrm{C}\mathrm{M}}=\ell }$ and ${\displaystyle I_O=m{\ell }^2}$, so the expression reduces to: ${\displaystyle {{\omega }_0}^2=\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}=\frac{mg\ell }{m{\ell }^2}=\frac{g}{\ell }}$ . Notice ${\displaystyle {\ell }^{\mathrm{e}\mathrm{q}}=\ell }$, as expected (the definition of equivalent length).
• A homogeneous rod of mass ${\displaystyle m}$ and length ${\displaystyle \ell }$ swinging from its end has ${\displaystyle r_{\mathrm{C}\mathrm{M}}=\frac{1}{2}\ell }$ and ${\displaystyle I_O=\frac{1}{3}m{\ell }^2}$, so the expression reduces to: ${\displaystyle {{\omega }_0}^2=\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}=\frac{mg\frac{1}{2}\ell }{\frac{1}{3}m{\ell }^2}=\frac{g}{\frac{2}{3}\ell }}$ . Notice ${\displaystyle {\ell }^{\mathrm{e}\mathrm{q}}=\frac{2}{3}\ell }$, a homogeneous rod oscilates as if it were a simple pendulum of two-thirds its length.
• A heavy simple pendulum: combination of a homogeneous rod of mass ${\displaystyle m_{\mathrm{r}\mathrm{o}\mathrm{d}}}$ and length ${\displaystyle \ell }$ swinging from its end, and a bob ${\displaystyle m_{\mathrm{b}\mathrm{o}\mathrm{b}}}$ at the other end. Then the system has a total mass of ${\displaystyle m_{\mathrm{b}\mathrm{o}\mathrm{b}}+m_{\mathrm{r}\mathrm{o}\mathrm{d}}}$, and the other parameters being ${\displaystyle r_{\mathrm{C}\mathrm{M}}=\frac{m_{\mathrm{b}\mathrm{o}\mathrm{b}}+m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}+2m_{\mathrm{r}\mathrm{o}\mathrm{d}}}\ell }$ and ${\displaystyle I_O=m_{\mathrm{b}\mathrm{o}\mathrm{b}}{\ell }^2+\frac{1}{3}{m}_{\mathrm{r}\mathrm{o}\mathrm{d}}{\ell }^2}$, so the expression reduces to:

${\displaystyle {{\omega }_0}^2=\frac{mg{r}_{\mathrm{C}\mathrm{M}}}{I_O}=\frac{(m_{\mathrm{b}\mathrm{o}\mathrm{b}}+m_{\mathrm{r}\mathrm{o}\mathrm{d}})g\frac{m_{\mathrm{b}\mathrm{o}\mathrm{b}}+m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}+2m_{\mathrm{r}\mathrm{o}\mathrm{d}}}\ell }{m_{\mathrm{b}\mathrm{o}\mathrm{b}}{\ell }^2+\frac{1}{3}{m}_{\mathrm{r}\mathrm{o}\mathrm{d}}{\ell }^2}=\frac{g}{\ell }\frac{(m_{\mathrm{b}\mathrm{o}\mathrm{b}}+m_{\mathrm{r}\mathrm{o}\mathrm{d}}{)}^2}{(m_{\mathrm{b}\mathrm{o}\mathrm{b}}+2m_{\mathrm{r}\mathrm{o}\mathrm{d}})\left(m_{\mathrm{b}\mathrm{o}\mathrm{b}}+\frac{1}{3}{m}_{\mathrm{r}\mathrm{o}\mathrm{d}}\right)}=\frac{g}{\ell }\frac{{\left(1+\frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}\right)}^2}{\left(1+2\frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}\right)\left(1+\frac{1}{3}\frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}\right)}}$. Notice this formula can be particularized into the two previous cases studied before just by considering the mass of the rod or the bob to be zero respectively. Also notice that the formula does not depend on both the mass of the bob and the rod, but on their ratio, ${\displaystyle \frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}}$. An approximation can be made for ${\displaystyle \frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}\ll 1}$: ${\displaystyle {{\omega }_0}^2\approx \frac{g}{\ell }\frac{1}{1+\frac{1}{3}\frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}}\approx \frac{g}{\ell }\left(1-\frac{1}{3}\frac{m_{\mathrm{r}\mathrm{o}\mathrm{d}}}{m_{\mathrm{b}\mathrm{o}\mathrm{b}}}+\cdots \right)}$

Notice how similar it is to the angular frequency in a spring-mass system with effective mass, and how Rayleigh's value appears.

Physical interpretation of the imaginary period

The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is, of course, the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: if θ₀ is the maximum angle of one pendulum and 180° − θ₀ is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.

Coupled pendula

Two identical simple pendulums coupled via a spring connecting the bobs.

Coupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in a supporting structure (such as a tabletop). The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian mechanics.

The kinetic energy of the system is:

$${\displaystyle E_{\text{K}}=\frac{1}{2}m{L}^2\left({\dot{\theta }}_1^2+{\dot{\theta }}_2^2\right)}$$

where ${\displaystyle m}$ is the mass of the bobs, ${\displaystyle L}$ is the length of the strings, and ${\displaystyle {\theta }_1}$, ${\displaystyle {\theta }_2}$ are the angular displacements of the two bobs from equilibrium.

The potential energy of the system is:

$${\displaystyle E_{\text{p}}=mgL(2-\cos {\theta }_1-\cos {\theta }_2)+\frac{1}{2}k{L}^2({\theta }_2-{\theta }_1{)}^2}$$

where ${\displaystyle g}$ is the gravitational acceleration, and ${\displaystyle k}$ is the spring constant. The displacement ${\displaystyle L({\theta }_2-{\theta }_1)}$ of the spring from its equilibrium position assumes the small angle approximation.

The Lagrangian is then

$${\displaystyle \mathcal{L}=\frac{1}{2}m{L}^2\left({\dot{\theta }}_1^2+{\dot{\theta }}_2^2\right)-mgL(2-\cos {\theta }_1-\cos {\theta }_2)-\frac{1}{2}k{L}^2({\theta }_2-{\theta }_1{)}^2}$$

which leads to the following set of coupled differential equations:

$${\displaystyle \begin{array}{rl}{\ddot{\theta }}_1+\frac{g}{L}\sin {\theta }_1+\frac{k}{m}({\theta }_1-{\theta }_2)& =0\\ {\ddot{\theta }}_2+\frac{g}{L}\sin {\theta }_2-\frac{k}{m}({\theta }_1-{\theta }_2)& =0\end{array}}$$

Adding and subtracting these two equations in turn, and applying the small angle approximation, gives two harmonic oscillator equations in the variables ${\displaystyle {\theta }_1+{\theta }_2}$ and ${\displaystyle {\theta }_1-{\theta }_2}$:

$${\displaystyle \begin{array}{rl}{\ddot{\theta }}_1+{\ddot{\theta }}_2+\frac{g}{L}({\theta }_1+{\theta }_2)& =0\\ {\ddot{\theta }}_1-{\ddot{\theta }}_2+\left(\frac{g}{L}+2\frac{k}{m}\right)({\theta }_1-{\theta }_2)& =0\end{array}}$$

with the corresponding solutions

$${\displaystyle \begin{array}{rl}{\theta }_1+{\theta }_2& =A\cos ({\omega }_{1}t+\alpha )\\ {\theta }_1-{\theta }_2& =B\cos ({\omega }_{2}t+\beta )\end{array}}$$

where

$${\displaystyle \begin{array}{rl}{\omega }_1& =\sqrt{\frac{g}{L}}\\ {\omega }_2& =\sqrt{\frac{g}{L}+2\frac{k}{m}}\end{array}}$$

and ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle \alpha }$, ${\displaystyle \beta }$ are constants of integration.

Expressing the solutions in terms of ${\displaystyle {\theta }_1}$ and ${\displaystyle {\theta }_2}$ alone:

$${\displaystyle \begin{array}{rl}{\theta }_1& =\frac{1}{2}A\cos ({\omega }_{1}t+\alpha )+\frac{1}{2}B\cos ({\omega }_{2}t+\beta )\\ {\theta }_2& =\frac{1}{2}A\cos ({\omega }_{1}t+\alpha )-\frac{1}{2}B\cos ({\omega }_{2}t+\beta )\end{array}}$$

If the bobs are not given an initial push, then the condition ${\displaystyle {\dot{\theta }}_1(0)={\dot{\theta }}_2(0)=0}$ requires ${\displaystyle \alpha =\beta =0}$, which gives (after some rearranging):

$${\displaystyle \begin{array}{rl}A& ={\theta }_1(0)+{\theta }_2(0)\\ B& ={\theta }_1(0)-{\theta }_2(0)\end{array}}$$

Small-angle approximation

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Small-angle_approximation

Approximately equal behavior of some (trigonometric) functions for x → 0

The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:

${\displaystyle \begin{array}{rl}\sin \theta & \approx \theta \\ \cos \theta & \approx 1-\frac{{\theta }^2}{2}\approx 1\\ \tan \theta & \approx \theta \end{array}}$

These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.

There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation, ${\displaystyle \cos \theta }$ is approximated as either ${\displaystyle 1}$ or as $1-\frac{{\theta }^2}{2}$.

Justifications

Graphic

The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.

• 
  Figure 1. A comparison of the basic odd trigonometric functions to θ. It is seen that as the angle approaches 0 the approximations become better.
• 
  Figure 2. A comparison of cos θ to 1 − θ²/2. It is seen that as the angle approaches 0 the approximation becomes better.

Geometric

The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the same length, meaning cos θ is close to 1 and θ²/2 helps trim the red away.

$${\displaystyle \cos \theta \approx 1-\frac{{\theta }^2}{2}}$$

The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = Aθ, from trigonometry, sin θ = O/H and tan θ = O/A, and from the picture, O ≈ s and H ≈ A leads to:

$${\displaystyle \sin \theta =\frac{O}{H}\approx \frac{O}{A}=\tan \theta =\frac{O}{A}\approx \frac{s}{A}=\frac{A\theta }{A}=\theta .}$$

Simplifying leaves,

$${\displaystyle \sin \theta \approx \tan \theta \approx \theta .}$$

Calculus

Using the squeeze theorem, we can prove that

$${\displaystyle \underset{\theta \to 0}{lim}\frac{\sin (\theta )}{\theta }=1,}$$

which is a formal restatement of the approximation ${\displaystyle \sin (\theta )\approx \theta }$ for small values of θ.

A more careful application of the squeeze theorem proves that

$${\displaystyle \underset{\theta \to 0}{lim}\frac{\tan (\theta )}{\theta }=1,}$$

from which we conclude that ${\displaystyle \tan (\theta )\approx \theta }$ for small values of θ.

Finally, L'Hôpital's rule tells us that

$${\displaystyle \underset{\theta \to 0}{lim}\frac{\cos (\theta )-1}{{\theta }^2}=\underset{\theta \to 0}{lim}\frac{-\sin (\theta )}{2\theta }=-\frac{1}{2},}$$

which rearranges to $\cos (\theta )\approx 1-\frac{{\theta }^2}{2}$ for small values of θ. Alternatively, we can use the double angle formula ${\displaystyle \cos 2A\equiv 1-2{\sin }^{2}A}$. By letting ${\displaystyle \theta =2A}$, we get that $\cos \theta =1-2{\sin }^2\frac{\theta }{2}\approx 1-\frac{{\theta }^2}{2}$.

Algebraic

The small-angle approximation for the sine function.

The Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is

$${\displaystyle \begin{array}{rl}\sin \theta & =\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{\theta }^{2n+1}\\ & =\theta -\frac{{\theta }^3}{3!}+\frac{{\theta }^5}{5!}-\frac{{\theta }^7}{7!}+\cdots \end{array}}$$

where θ is the angle in radians. In clearer terms,

$${\displaystyle \sin \theta =\theta -\frac{{\theta }^3}{6}+\frac{{\theta }^5}{120}-\frac{{\theta }^7}{5040}+\cdots }$$

It is readily seen that the second most significant (third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of 0.000001, or 1/10000 the first term. One can thus safely approximate:

$${\displaystyle \sin \theta \approx \theta }$$

By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine,

$${\displaystyle \tan \theta \approx \sin \theta \approx \theta ,}$$

Dual numbers

One may use dual numbers, denoted ε, often considered similar to an infinitesimal amount, with a square of 0. By using the Maclaurin series of cosine and sine and substituting θ=θε, the result is cos(θε)=1 and sin(θε)=θε. These approximations satisfy the Pythagorean identity:

cos²(θε)+sin²(θε)=1²+(θε)²=1+θ²ε²=1+θ²0=1.

Error of the approximations

Figure 3. A graph of the relative errors for the small angle approximations.

Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:

• cos θ ≈ 1 at about 0.1408 radians (8.07°)
• tan θ ≈ θ at about 0.1730 radians (9.91°)
• sin θ ≈ θ at about 0.2441 radians (13.99°)
• cos θ ≈ 1 − θ²/2 at about 0.6620 radians (37.93°)

Angle sum and difference

The angle addition and subtraction theorems reduce to the following when one of the angles is small (β ≈ 0):

cos(α + β)	≈ cos(α) − β sin(α),
cos(α − β)	≈ cos(α) + β sin(α),
sin(α + β)	≈ sin(α) + β cos(α),
sin(α − β)	≈ sin(α) − β cos(α).

Specific uses

Astronomy

In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation. The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:

${\displaystyle D=X\frac{d}{206265{}^{″}}}$

where X is measured in arcseconds.

The quantity 206265″ is approximately equal to the number of arcseconds in a circle (1296000″), divided by 2π, or, the number of arcseconds in 1 radian.

The exact formula is

${\displaystyle D=d\tan \left(X\frac{2\pi }{1296000{}^{″}}\right)}$

and the above approximation follows when tan X is replaced by X.

Motion of a pendulum

The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion.

When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.

Optics

In optics, the small-angle approximations form the basis of the paraxial approximation.

Wave Interference

The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen, and d is the distance between the slits: 

$${\displaystyle y\approx \frac{m\lambda D}{d}}$$

Structural mechanics

The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling). This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.

Piloting

The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.

Interpolation

The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values:

Example: sin(0.755)

$${\displaystyle \begin{array}{rl}\sin (0.755)& =\sin (0.75+0.005)\\ & \approx \sin (0.75)+(0.005)\cos (0.75)\\ & \approx (0.6816)+(0.005)(0.7317)\\ & \approx \mathrm{0.6853.}\end{array}}$$

where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table.

Uses of trigonometry

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Uses_of_trigonometry

Trigonometry

The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles.

Amongst the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.

Thomas Paine's statement

In Chapter XI of The Age of Reason, the American revolutionary and Enlightenment thinker Thomas Paine wrote:

The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of any thing else relating to the motion of the heavenly bodies, are contained chiefly in that part of science that is called trigonometry, or the properties of a triangle, which, when applied to the study of the heavenly bodies, is called astronomy; when applied to direct the course of a ship on the ocean, it is called navigation; when applied to the construction of figures drawn by a ruler and compass, it is called geometry; when applied to the construction of plans of edifices, it is called architecture; when applied to the measurement of any portion of the surface of the earth, it is called land-surveying. In fine, it is the soul of science. It is an eternal truth: it contains the mathematical demonstration of which man speaks, and the extent of its uses are unknown.

History

Great Trigonometrical Survey

Main article: Great Trigonometrical Survey

From 1802 until 1871, the Great Trigonometrical Survey was a project to survey the Indian subcontinent with high precision. Starting from the coastal baseline, mathematicians and geographers triangulated vast distances across the country. One of the key achievements was measuring the height of Himalayan mountains, and determining that Mount Everest is the highest point on Earth. 

Historical use for multiplication

For the 25 years preceding the invention of the logarithm in 1614, prosthaphaeresis was the only known generally applicable way of approximating products quickly. It used the identities for the trigonometric functions of sums and differences of angles in terms of the products of trigonometric functions of those angles.

Some modern uses

Scientific fields that make use of trigonometry include:

acoustics, architecture, astronomy, cartography, civil engineering, geophysics, crystallography, electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, machining, medical imaging, number theory, oceanography, optics, pharmacology, probability theory, seismology, statistics, and visual perception

That these fields involve trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It does mean that some things in these fields cannot be understood without trigonometry. For example, a professor of music may perhaps know nothing of mathematics, but would probably know that Pythagoras was the earliest known contributor to the mathematical theory of music.

In some of the fields of endeavor listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. In the case of music theory, the application of trigonometry is related to work begun by Pythagoras, who observed that the sounds made by plucking two strings of different lengths are consonant if both lengths are small integer multiples of a common length. The resemblance between the shape of a vibrating string and the graph of the sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some waves and the graph of the sine function is also not coincidental. In some other fields, among them climatology, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine functions.

Fourier series

Many fields make use of trigonometry in more advanced ways than can be discussed in a single article. Often those involve what are called the Fourier series, after the 18th- and 19th-century French mathematician and physicist Joseph Fourier. Fourier series have a surprisingly diverse array of applications in many scientific fields, in particular in all of the phenomena involving seasonal periodicities mentioned above, and in wave motion, and hence in the study of radiation, of acoustics, of seismology, of modulation of radio waves in electronics, and of electric power engineering.

A Fourier series is a sum of this form:

${\displaystyle ◻+\underset{1}{\underbrace{◻\cos \theta +◻\sin \theta }}+\underset{2}{\underbrace{◻\cos (2\theta )+◻\sin (2\theta )}}+\underset{3}{\underbrace{◻\cos (3\theta )+◻\sin (3\theta )}}+\cdots }$

where each of the squares (${\displaystyle ◻}$) is a different number, and one is adding infinitely many terms. Fourier used these for studying heat flow and diffusion (diffusion is the process whereby, when you drop a sugar cube into a gallon of water, the sugar gradually spreads through the water, a pollutant spreads through the air, or any dissolved substance spreads through any fluid).

Fourier series are also applicable to subjects whose connection with wave motion is far from obvious. One ubiquitous example is digital compression whereby images, audio and video data are compressed into a much smaller size which makes their transmission feasible over telephone, internet and broadcast networks. Another example, mentioned above, is diffusion. Among others are: the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenberg's inequality.

Fourier transforms

A more abstract concept than Fourier series is the idea of Fourier transform. Fourier transforms involve integrals rather than sums, and are used in a similarly diverse array of scientific fields. Many natural laws are expressed by relating rates of change of quantities to the quantities themselves. For example: The rate population change is sometimes jointly proportional to (1) the present population and (2) the amount by which the present population falls short of the carrying capacity. This kind of relationship is called a differential equation. If, given this information, one tries to express population as a function of time, one is trying to "solve" the differential equation. Fourier transforms may be used to convert some differential equations to algebraic equations for which methods of solving them are known. Fourier transforms have many uses. In almost any scientific context in which the words spectrum, harmonic, or resonance are encountered, Fourier transforms or Fourier series are nearby.

Statistics, including mathematical psychology

Intelligence quotients are sometimes held to be distributed according to a bell-shaped curve. About 40% of the area under the curve is in the interval from 100 to 120; correspondingly, about 40% of the population scores between 100 and 120 on IQ tests. Nearly 9% of the area under the curve is in the interval from 120 to 140; correspondingly, about 9% of the population scores between 120 and 140 on IQ tests, etc. Similarly many other things are distributed according to the "bell-shaped curve", including measurement errors in many physical measurements. Why the ubiquity of the "bell-shaped curve"? There is a theoretical reason for this, and it involves Fourier transforms and hence trigonometric functions. That is one of a variety of applications of Fourier transforms to statistics.

Trigonometric functions are also applied when statisticians study seasonal periodicities, which are often represented by Fourier series.

Number theory

There is a hint of a connection between trigonometry and number theory. Loosely speaking, one could say that number theory deals with qualitative properties rather than quantitative properties of numbers.

${\displaystyle \frac{1}{42},\frac{2}{42},\frac{3}{42},\dots \dots ,\frac{39}{42},\frac{40}{42},\frac{41}{42}.}$

Discard the ones that are not in lowest terms; keep only those that are in lowest terms:

${\displaystyle \frac{1}{42},\frac{5}{42},\frac{11}{42},\dots ,\frac{31}{42},\frac{37}{42},\frac{41}{42}.}$

Then bring in trigonometry:

${\displaystyle \cos \left(2\pi \cdot \frac{1}{42}\right)+\cos \left(2\pi \cdot \frac{5}{42}\right)+\cdots +\cos \left(2\pi \cdot \frac{37}{42}\right)+\cos \left(2\pi \cdot \frac{41}{42}\right)}$

The value of the sum is −1, because 42 has an odd number of prime factors and none of them is repeated: 42 = 2 × 3 × 7. (If there had been an even number of non-repeated factors then the sum would have been 1; if there had been any repeated prime factors (e.g., 60 = 2 × 2 × 3 × 5) then the sum would have been 0; the sum is the Möbius function evaluated at 42.) This hints at the possibility of applying Fourier analysis to number theory.

Solving non-trigonometric equations

Various types of equations can be solved using trigonometry.

For example, a linear difference equation or linear differential equation with constant coefficients has solutions expressed in terms of the eigenvalues of its characteristic equation; if some of the eigenvalues are complex, the complex terms can be replaced by trigonometric functions of real terms, showing that the dynamic variable exhibits oscillations.

Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers; again an alternative solution exists in terms of trigonometric functions of real terms.