In geometry, trigonometry often means right triangles: opposite, adjacent, hypotenuse. In precalculus, the same ratios open up into the unit circle and into periodic graphs that model tides, sound, and rotation.
The unit circle idea
Draw a circle of radius 1 centered at the origin. Measure an angle from the positive x-axis. The point where the ray hits the circle has coordinates:
- (cos θ, sin θ)
- So cosine is the left-right position
- Sine is the up-down position
That is why sin(90°) = 1 (top of the circle) and cos(0°) = 1 (rightmost point). You do not have to memorize those as random facts once the picture is clear.
Degrees and radians
A full turn is 360° or 2π radians. Radians measure angle by arc length on the unit circle: an angle of 1 radian cuts an arc of length 1. Calculus prefers radians; precalculus trains you to convert:
180° = π radians, so 1° = π/180 and 1 radian = 180/π degrees.
Sine and cosine as waves
Plot angle on the horizontal axis and sine on the vertical axis. As the angle runs forward, the height goes up and down—producing the familiar wave. Important features:
- Amplitude — height of the wave from the middle to a peak (for y = A sin x, amplitude is |A|)
- Period — how long one full cycle takes (for sin x, period is 2π)
- Midline — horizontal center line after a vertical shift
Tangent and ratios
tan θ = sin θ / cos θ when cosine is not zero. On a ramp, tangent is rise over run—the slope. At 45° (π/4 radians), rise equals run, so tangent equals 1.
Practice: Standard → Trigonometry uses plain-language scenarios (ladders, waves, tides) with graphs. Honors adds identities and applications. Start free practice.