Trigonometric Functions

This page explains trigonometric functions, a very important concept in measurement, and their applications.

<Note>
This page uses MathJax to display formulas. If your browser settings block script execution, the formulas on this page will not display correctly.

Trigonometric function basics

Trigonometric functions, also called circular functions, is the collective name for functions expressing the relationship between the size of angles and the lengths of line segments in a triangle on a plane, and the functions obtained by extending them. The trigonometric functions comprise six functions, called as follows.

Each function is expressed as follows using the unit circle (the circle centered at the origin with radius 1):

Definition
Take an arbitrary point \(\ \mathrm{P} (x,y)\ \) on the unit circle, and let \(\ \theta\ \) be the angle between the ray \(\ \mathrm{OP}\ \) and the \(\ x\ \)-axis (angles are measured counterclockwise). Then

The following relations also hold between these functions.

Theorem
For any angles \(\ x\ \) and \(\ y\ \), the following equations always hold.

From their definition, you can see that the trigonometric functions are periodic functions, in which the same values appear periodically.
The period of the trigonometric functions is 360 degrees for sine, cosine, secant and cosecant, and 180 degrees for tangent and cotangent.

Below are the values of some trigonometric functions for simple angles.

Note: Because the denominator becomes 0, \(\tan 90^\circ,\ \tan 270^\circ, \dots,\ \cot 0^\circ,\ \cot 180^\circ, \dots\ \) and so on are undefined.

Radians and the 256-direction system

When we normally measure angles, we use degrees, with a right angle being \(\ 90^\circ\ \) and a full turn \(\ 360^\circ\ \), but mathematically it is often more convenient to express angles another way.
That is the measure called radians, in which a full turn is \(\ 2\pi\ \mathrm{rad}\ \). The unit symbol \(\ \mathrm{rad}\ \) is read “radian”, and it is usually omitted when it is clear that radians are being used. On this page as well, angles are in radians from here on unless otherwise stated.
Note that \(\ \pi\ \) is pi, the ratio of a circle’s circumference to its diameter.

Degrees and radians correspond as follows.

Theorem
Between the value \(\ \theta\ \) of an angle measured in degrees and the value \(\ \phi \ \) of the same angle measured in radians, the following equation always holds.

In the SDK series, a 256-direction system in which a full turn is \(\ 256\mbox{^}\ \) is used as the unit of angle.
Because the values representable in 8 bits are 0 to 255, and because a computer’s internal arithmetic is basically binary, this is easier to use than degrees and the like.
Since it is not a standard unit of measure, it has no particular name or unit, but for convenience this site uses \(\ \mbox{^}\ \) as its unit symbol.
So, for example, \(\ 45^\circ=\cfrac{\pi}{4}\ \mathrm{rad}=32\mbox{^}\ \).
Since the values of trigonometric functions are fractions from \(\ -1\ \) to \(\ 1\ \), games generally keep them as tables of fixed-point numbers (for example, in 8.8 format, treating 1 as 256).

Computing trigonometric functions

Trigonometric functions have various important properties, one of which is that they can be computed using relationships between trigonometric functions.
These are basic and convenient aspects of trigonometric functions, so be sure to remember them.

When you want to find an unknown trigonometric value, the following angle addition formulas let you compute it using known trigonometric values.

Theorem (angle addition formulas)
For any angles \(\ x\ \) and \(\ y\ \), the following equations always hold.

Using the angle addition formulas, the following multiple-angle and half-angle formulas are derived.

Theorem (double-angle formulas)
For any angle \(\ x\ \), the following equations always hold.

Theorem (triple-angle formulas)
For any angle \(\ x\ \), the following equations always hold.

Theorem (half-angle formulas)
For any angle \(\ x\ \), the following equations always hold.

Also, using de Moivre’s theorem or Euler’s formula for exponential functions, more general multiple-angle and fractional-angle formulas can be written as follows using the imaginary unit \(\ i\ \).

Theorem (n-fold angle formulas)
For any angle \(\ x\ \), the following equations always hold.

The following relations for sums and products of sine and cosine are derived from the angle addition formulas.

Theorem (sum-to-product and product-to-sum formulas)
For any angles \(\ x,y\ \), the following equations always hold.

For the sum of a sine and a cosine, there is the following formula.

Theorem (harmonic addition formula)
For any angle \(\ x\ \) and suitable constants \(\ a,b\ \), the following equation always holds.

For \(\arcsin\ \), see the section on inverse trigonometric functions at the bottom of this page.

Applications of trigonometric functions (1): measuring lengths

Trigonometric functions have been used for thousands of years since ancient Egypt, originally out of the need to survey land.
From their history of use in surveying (where they are still in active use and will remain so into the future), they are known to have many geometric and mathematical properties, and their uses are wide-ranging.
Apart from their original purpose of geometric measurement, they also have physically very important properties (for example, it is well known that the waveform produced when a tuning fork is struck is a sine wave), and the next section covers circular motion and simple harmonic motion as a very familiar example. This section explains very basic content related to measurement.

If you have a triangle and know the lengths of two sides and the angle between those two sides, you can find the area of the triangle.
More precisely, if you know any of
(1) the lengths of the three sides
(2) the lengths of two sides and the size of the angle between them
(3) the length of one side and the sizes of the angles at both its ends
then you can use the various theorems described below to find the remaining side lengths and angles and the area of the triangle. This is called solving the triangle.
These conditions are exactly the same as the triangle congruence conditions, that is, they are also the conditions that uniquely determine the shape of a triangle (similarity conditions, such as all angles being known, do not determine the size, so the side lengths cannot be found).

Theorem
In any \(\ \triangle\mathrm{ABC}\ \), the area \(\ S\ \) of the triangle can be written as follows using the lengths of its sides and the angles at its vertices.

Using trigonometric functions, the Pythagorean theorem for right triangles can be extended so that it applies to any triangle.

Theorem (first law of cosines)
In any \(\ \triangle\mathrm{ABC}\ \), the following relations hold between the lengths of its sides and the angles at its vertices.

Using the first law of cosines, the following equations are derived.

Theorem (second law of cosines)
In any \(\ \triangle\mathrm{ABC}\ \), the following relations hold between the lengths of its sides and the angles at its vertices.

For a right triangle, one of the cosines in the second law of cosines becomes 0.
Looking at the equation in which the cosine is 0, that equation is exactly the Pythagorean theorem. This is why the law of cosines is called the generalized Pythagorean theorem.
Also, solving the second law of cosines for the cosines gives the following equations. This means that if the lengths of the three sides of a triangle are known, all the vertex angles can be found from them.

Theorem
In any \(\ \triangle\mathrm{ABC}\ \), the cosine of the angle at each vertex can be written as follows using the side lengths of the triangle.

There is also the following theorem concerning the circumcircle of a triangle.

Theorem (law of sines)
In any \(\ \triangle\mathrm{ABC}\ \), the radius \(\ R\ \) of the triangle’s circumcircle can be written as follows using the side lengths and vertex angles of the triangle.

Using this theorem, you can find the lengths of the remaining two sides from one side and the angles at both its ends.

If you want to know the area of a regular polygon, there is the following formula.

Theorem
For any regular \(\ n\ \)-gon, its area \(\ S\ \) can be written as follows, where \(\ a\ \) is the length of one side of the polygon.

Applications of trigonometric functions (2): circular motion and simple harmonic motion

The reason trigonometric functions are also called circular functions is that they can themselves be defined using a circle. Indeed, at the beginning of this page, we defined the trigonometric functions using the unit circle. Trigonometric functions are known to be definable in various other ways too, but whichever definition is used, the resulting trigonometric functions have exactly the same properties and values.
One physical phenomenon closely related to trigonometric functions is wave motion, and analyzing waves in detail leads to the physical phenomenon called simple harmonic motion. This simple harmonic motion is closely related to uniform circular motion (literally, motion around a circle at a constant speed). If trigonometric functions can be defined using a circle, circular motion should also be describable using trigonometric functions. Going deep into physics would stray from the purpose of this page, so only the bare minimum is explained briefly.

By the unit-circle definition of the trigonometric functions, if we define a point \(\ \mathrm{P}\ \) on the \(\ xy\ \)-plane using a parameter \(\ \phi\ \) as

then the locus traced by this point \(\ \mathrm{P}\ \) is a circle of radius \(\ r\ \) centered at the origin.
Now, when this \(\ \phi\ \) can be written as \(\ \phi=\omega t+\alpha\ \) using time \(\ t\ \) and constants \(\ \alpha,\ \omega\ \), the point moves around the circumference at a constant speed. This is uniform circular motion.
In this uniform circular motion, \(\ \omega\ \) is called the angular velocity and \(\ \alpha\ \) the initial phase (the angle at which point \(\ \mathrm{P}\ \) was initially located).
The time it takes for point \(\ \mathrm{P}\ \) to go exactly once around the circumference, that is, the period of this circular motion, is \(\ T=\cfrac{2\pi}{\omega}\ \), and the frequency (number of revolutions per unit time) is \(\ f=\cfrac{\omega}{2\pi}=\cfrac{1}{T}\ \).
Also, when rotating at the same angular velocity, the farther a point is from the center of rotation, the faster it moves. The speed at radius \(\ r\ \) is \(\ v=r\omega \ \).

Now, let’s view the circular motion not from directly above (perpendicular to the plane of rotation) but from the side (parallel to the plane of rotation). How does the motion of point \(\ \mathrm{P}\ \) look?
If you actually try it, it’s easy to see. It oscillates vertically. This oscillation is the same as the motion called simple harmonic motion (free oscillation).
A simple example of simple harmonic motion is the oscillation of a weight hung from a spring. It is a common motion all around us.
Since simple harmonic motion is exactly identical to the orthogonal projection of uniform circular motion (the motion of its shadow when light is shone parallel to the plane of rotation), the properties of simple harmonic motion are very similar to those of uniform circular motion.
The period is \(\ T=\cfrac{2\pi}{\omega}\ \), the same as for circular motion, and the frequency also matches the frequency (revolutions) of the circular motion, \(\ f=\cfrac{\omega}{2\pi}=\cfrac{1}{T}\ \). The radius \(\ r\ \) of the circular motion corresponds to the amplitude \(\ A\ \) of this oscillation, and the initial phase of the circular motion becomes the initial phase of the simple harmonic motion as is. The coordinate at time \(\ t\ \) is \(\ x=A\sin(\omega t+\alpha)\ \).
Since simple harmonic motion is a projection of circular motion, for example, if you stop updating one of the \(\ x,\ y\ \) coordinates of a Zinger moving in a circle, that Zinger will oscillate vertically or horizontally in simple harmonic motion (incidentally, the back-and-forth motion in the original SDK series games is uniform linear motion except at the turnaround ends).
By understanding physical phenomena and the mathematics hidden behind them in this way, you become able to introduce new movements not found in the original games.

Inverse trigonometric functions

Functions that find the original angle from the value of a trigonometric function are called inverse trigonometric functions. However, because of the periodicity of trigonometric functions, multiple angles would result as is. If this is inconvenient, the possible values can be restricted to narrow it down to one. In mathematics this is actually the usual practice, and all of those listed below are the so-called principal values. Of course, you may use values other than the principal values if you wish, but usually using this definition is sufficient.

Since notations like \(\ \arcsin x\ \) are long, \(\ \sin^{-1} x\ \) or \(\ \mathrm{Sin}^{-1} x\ \) are sometimes written instead.
\(\ \sin^{-1} x\ \) does not mean sine to the power of minus 1, i.e. \(\cfrac{1}{\sin x}\ \). The same goes for the others.

Below are the values of some inverse trigonometric functions for simple values.

Note: For angles at which \(\ \tan\ \), \(\ \cot\ \) and so on are undefined, the corresponding inverse trigonometric functions are also undefined.

There is also the following relation for inverse trigonometric functions.

Theorem
For any \(\ x\ \) at which the arcsine and arccosine are defined, always \(\ \arcsin x+\arccos x=\cfrac{\pi}{2}\ \)

Trigonometric tables

Exact values of commonly used trigonometric functions, and trigonometric tables in degrees and in the 256-direction system (to 4 decimal places), are given below.
The tables in degrees and in the 256-direction system list only values from 0 to 90 degrees, but values for other angles are easily found from the relationships between trigonometric functions.

Radians

Degrees

256 directions