Knowledge Node - Trigonometry

Trigonometry literally means the measure of triangles. Yet, I believe that to fundamentally understand trigonometry, one must understand the unit circle. In mathematics, unit means 1. The Unit Circle is a circle centered at the origin of radius 1.

The cosine, sine, and tangent functions of Trigonometry are all based off of triangles made with the unit circle. Imagine a line starting at the origin and extending to the coordinate x = +1, y = 0. This line then sweeps counter-clockwise. The angle (commonly called theta) is measured between the positive x-axis and the imaginary line. The triangle created by this sweeping imaginary line always has two points on the x-axis. The third point exists on the circle.

The cosine function is the horizontal leg of the triangle. The sine function is the vertical leg of the triangle. The tangent function is the vertical leg divided by the horizontal leg, or the slope of the hypotenuse. The sine and cosine functions repeat every 360 degrees, and the tangent function repeats every 180 degrees.

Radians are an alternative to degrees. A circle consists of radians or 360 degrees. The radian measurement is equal to the arc-length of the curve drawn by the given angle. For example, the angle radians is both the circumference of the circle and the equivalent to 360 degrees.

Note: A good programming algorithm to know is the conversion from degrees to radians. Since OpenGL uses degrees these functions can useful tools.

float D_TO_R( float x) { return PI / 180.0f * x; }
float R_TO_D( float x) { return 180.0f * x / PI; }

The cosine curve, once again, is the horizontal measured distance from the origin to the endpoint on the x-axis. Notice how it initially begins at +1 and moves to 0 and then -1. This is mirrored by the imaginary line pointing to the right, sweeping counterclockwise upwards and then to the left.

The sine curve, is the vertical distance from the x-axis to the last point on the circle. Notice that it begins at 0, and as the horizontal line moves counterclockwise away from the positive x-axis, the vertical distance increases, peaks, and decreases again.

The vertical lines on the tangent graph are vertical asymptotes. Since the graph is defined as the Sine curve divided by the Cosine curve, whenever the Cosine curve is zero, a vertical asymptote appears on the tangent curve. This is also mirrored in the fact that the tangent curve is the slope of the horizontal line, and whenever the horizontal line is pointing either straight up or straight down, it's slope becomes infinity.