Trigonometric Functions
Prerequisites Required- Quadratic Functions
Previously, we discussed linear and quadratic functions. Amongst the landscape of mathematical functions, trigonometric functions are some of the most important, having critical applications in geometry, physics, and engineering. To be honest, the study of triangles to such an intense degree seemed absurd to me until later when I realized that trigonometric functions are used everywhere!
Angles
Angle is the amount of rotation between the two lines; the amount one line would have to rotate to be aligned in the same direction as the other.
We don't normally represent angle in units of rotation but instead in units of degrees or radians. Angles in degrees are often notated with a small circle, $120$ degrees is written $120^\circ$.
Rotations to Degrees: $\theta (degrees) = \theta (rotations) *360$
$360$ degrees being equal to $1$ rotation is mostly historical baggage from ancient Babylonian astronomy and Greek geometry. However, it's very pervasive.
As an alternative to rotations or degrees, we can instead measure an angle based on the length of the radius $1$ arc that subtends it, with units of radians. The circumference of a circle of radius $1$ is $2\pi$ which is equal to $6.2831853072...$. The nature of $\pi$ is described in more detail in the below article.
Jonathan Bowen
Consider the below triangle/arc that is $\frac{1}{32}$ of a rotation.
$\frac{1}{32}$ rotation corresponds to an angle of $11.25^\circ$.
The length of the radius $1$ arc created by this angle is equal to the radius $1$ circle's circumference divided by $32$:
$$L = 2\pi\frac{1}{32} = \frac{\pi}{16} = 0.1963495408...$$
Therefore, rather than calling this angle $\frac{1}{32}$ of a rotation or $11.25^\circ$, we say this angle is $\frac{\pi}{16}$ radians.
Similar to degrees, there is a straightforward conversion from rotations to radians:
Rotations to Radians: $\theta (radians) = \theta (rotations) *2\pi$
Given the above angles, we can convert rotations to degrees and radians:
$3/8\:rotation = 135^\circ = \frac{3\pi}{4}\:radians$
$1/4\:rotation = 90^\circ = \frac{\pi}{2}\:radians$
$1/8\:rotation = 45^\circ = \frac{\pi}{4}\:radians$
$1/16\:rotation = 22.5^\circ = \frac{\pi}{8}\:radians$
$1/32\:rotation = 11.25^\circ = \frac{\pi}{16}\:radians$
Radians are the more standard measure for angle, but because degrees are so common, we will utilize both units regularly in this article.
The Right Triangle
Our adventure in trigonometry begins with the right triangle. A triangle is a shape with three sides, and a right triangle is a triangle that has a $90^\circ$ angle.