Unit Circle

The unit circle extends trigonometry beyond right triangles to all real numbers, allowing us to define sine and cosine for angles greater than $90^{\circ }$ and negative angles.

Definition

For any angle $\theta$ (measured counterclockwise from the positive x-axis):

• $x\text{-coordinate}=\cos \theta$
• $y\text{-coordinate}=\sin \theta$

Radians vs Degrees

Olympiad mathematics almost exclusively uses radians.

• $360^{\circ }=2\pi$ radians
• $180^{\circ }=\pi$ radians
• Conversion: $\text{radians}=\text{degrees}\times \frac{\pi }{180}$.

ASTC Rule (All Students Take Calculus)

This mnemonic tells you which functions are positive in which quadrant:

• Quadrant I ($0$ to $\pi /2$): All are positive.
• Quadrant II ($\pi /2$ to $\pi$): Sine is positive.
• Quadrant III ($\pi$ to $3\pi /2$): Tangent is positive.
• Quadrant IV ($3\pi /2$ to $2\pi$): Cosine is positive.

Common Values

$\sin \theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan \theta$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	Undef

Periodicity and Symmetry

Theorem (Periodic Properties)

* $\sin (\theta +2\pi )=\sin \theta$

• $\cos (\theta +2\pi )=\cos \theta$
• $\tan (\theta +\pi )=\tan \theta$ (Period is $\pi$)

Theorem (Negative Angles)

* $\cos (-\theta )=\cos \theta$ (Even function)

• $\sin (-\theta )=-\sin \theta$ (Odd function)

Practice Problems

Exercise (Problem 1)

Evaluate without a calculator: $\sin (\frac{7\pi }{6})\cdot \cos (\frac{5\pi }{3})$.

Exercise (Problem 2)

Find all $\theta$ in $[0,2\pi ]$ such that $\sin \theta =-\frac{1}{2}$.

Exercise (Problem 3)

Prove that points $A(\cos \alpha ,\sin \alpha )$, $B(\cos \beta ,\sin \beta )$, and $C(1,0)$ form an isosceles triangle if $\alpha =-\beta$.