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The Unit Circle

Extend trigonometry beyond right triangles

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The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. It's fundamental to understanding trigonometry.

Definition on Unit Circle

For any angle $\theta$ , if the terminal side intersects the unit circle at point $(x, y)$ :

$\cos\theta = x$ $\sin\theta = y$

Why the Unit Circle?

Extends trig functions to all angles (not just 0° to 90°)
Reveals patterns and symmetry
Makes graphing trig functions intuitive

Key Points

Angle (radians)	Angle (degrees)	(cos, sin)
0	0°	(1, 0)
$\frac{\pi}{6}$	30°	$(\frac{\sqrt{3}}{2}, \frac{1}{2})$
$\frac{\pi}{4}$	45°	$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$
$\frac{\pi}{3}$	60°	$(\frac{1}{2}, \frac{\sqrt{3}}{2})$
$\frac{\pi}{2}$	90°	(0, 1)
$\pi$	180°	(-1, 0)
$\frac{3\pi}{2}$	270°	(0, -1)
$2\pi$	360°	(1, 0)

Quadrant Signs

Quadrant	sin	cos	tan
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

Mnemonic: "All Students Take Calculus"

All positive in Q1
Sin positive in Q2
Tan positive in Q3
Cos positive in Q4

Reference Angles

The reference angle is the acute angle formed with the x-axis.

Use reference angles to find trig values in any quadrant:

Find the reference angle
Evaluate the trig function at the reference angle
Apply the correct sign based on the quadrant

Introduction to Trigonometry

Trigonometric Identities

CramClub • The Unit Circle | Pre-Calculus | CramClub