Trigonometric Functions
Key points
Radians: Measuring Angles by Arc Length
Instead of splitting a circle into 360 arbitrary pieces, a radian measures an angle by how far you travel along the circle's edge. Half a turn is π radians, so 180° = π, 90° = π/2, and a full turn 360° = 2π. Degrees are fine for everyday talk, but every physics formula and every programming language uses radians by default.
Defining sin and cos on the Unit Circle
Draw a circle of radius 1 centered at the origin and walk around it by an angle θ. Wherever you land, the x-coordinate is cos θ and the y-coordinate is sin θ. This definition is the key upgrade: a triangle only has angles under 90°, but you can keep walking around the circle forever, so sin and cos now work for 400°, for 1000°, and for negative angles too.
The Addition Formulas
This tells you sin of a sum of two angles in terms of the separate angles. It is the single formula worth knowing best, because the double-angle and half-angle formulas are all just special cases of it. Signal processing, optics, and quantum mechanics lean on it constantly.
The Double-Angle Formulas
Set α = β in the addition formula and it collapses into these. You don't have to memorize them separately — derive them in ten seconds when you need them. They sit underneath the rotation matrices used in computer graphics.
Amplitude, Frequency, and Phase
Four numbers describe any wave: y = A sin(Bx + C) + D. A is the amplitude — how loud the sound is. B is the angular frequency — how high the pitch is. C is the phase — how far the wave is shifted sideways. D just lifts the whole wave up or down. Every song you've heard and every radio signal you've used is a stack of these, added together.
Jobs that use this
Audio Engineer$82k
Communications Systems Engineer$108k
MRI Technologist / Medical Physicist$95k
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