Lesson 9.5

Trigonometric Functions on the Unit Circle

Triangles are limited to 180 degrees. Cycles go on forever. It's time to trap the triangle inside a circle.

Introduction

SOH CAH TOA is great for building ramps, but what if the angle is ? Only a circle can handle angles larger than 90. We define a standard circle with radius (the Unit Circle) to generalize sine and cosine.

Prerequisite Connection

Since , the Hypotenuse is always 1. Thus, Sine (Opp/Hyp) just becomes Opposite/1 (the y-coordinate).

Today's Increment

We redefine Trig Functions: and . This works for ANY angle, even negative ones.

Why This Matters

This is the definition of a Periodic Function. In Calculus, we integrate sine and cosine over periods. Without the unit circle definition, things like "Alternating Current" (waves) wouldn't make sense mathematically.

Key Concepts

Interactive Unit Circle

Drag the Point

r = 1

Current Angle

45°

Primary Functions

sin(θ)

cos(θ)

tan(θ)

Reciprocal Functions

csc

sec

cot

The Unit Circle Definition

Let be an angle in standard position. Its terminal side intersects the unit circle () at a unique point . As you move the point above, notice:

x= ("Horizontal Distance")
y= ("Vertical Distance")

Worked Examples

Example 1: Quadrantal Angles

Find and .

Visualize Angle

radians is . It points to the far left of the circle.

Identify Coordinator

The coordinates on the far left of the unit circle are .

(x-coord)

(y-coord)

Example 2: Finding Trig Values from a Point

The point is on the unit circle. Find .

Identify Sine and Cosine

.
.

Calculate Tangent

Example 3: Using Identity

If and is in Quadrant II, find .

Use Pythagorean Identity

Determine Sign

. But is it positive or negative?
In Quadrant II, x-values (Cosine) are negative.

Common Pitfalls

Slope Confusion

Students forget that Tangent is literally the Slope of the radius line. If the line goes up, Tan is positive. If it goes down, Tan is negative.

Real-World Application

Signal Processing

Your phone converts sound waves into digital signals using Fourier geometry, which breaks down any complex wave into a sum of simple sines and cosines on the unit circle. The x and y coordinates literally represent the phase and amplitude of the signal.