Sine and Cosine Definition

Pre-Calculus · Axiom Academy

LESSON Sine and Cosine Definitions The two functions that turn going around a circle into a wave — cos θ is where you are sideways, sin θ is how high. Drop a point on the unit circle at angle θ . Reading straight down to the x-axis gives a horizontal length — that's . Reading straight across to the y-axis gives a vertical length — that's . They aren't separate formulas to memorize; they are the coordinates (x, y) of the point. cos θ — the x-coordinate (horizontal) sin θ — the y-coordinate (vertical) As θ grows from 0 to , the point's x-coordinate slides between 1 and -1 . Plot that x-value against the angle and it paints out a smooth curve: the cosine wave . Same point, same trip around the circle — but now track its height . Transfer that height to a plot against θ and you get the sine wave , identical in shape to cosine but shifted: it starts at 0 , not 1 . Since cosine and sine are just the x and y coordinates, their signs are decided by which quadrant the point lands in. Watch the horizontal leg flip blue↔gray and the vertical leg flip orange↔gray as the point crosses each axis. Right and up: , . Both positive. Left and down: , . Both negative. A handful of angles come up constantly. Watch the point snap to each landmark and read its exact pair straight off the circle. You've seen sine and cosine not as mystery buttons on a calculator but as the plain coordinates of a point going around a circle. Scroll up to revisit any step.

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