Trigonometry Essentials

The unit circle

The unit circle is the circle of radius 1 centered at the origin. For any angle $\theta$ measured counterclockwise from the positive x-axis, the point on the circle is:

$(\cos \theta, \sin \theta)$

That's the cleanest definition of sine and cosine — they're just the coordinates of a point spinning around the unit circle.

Radians vs. degrees

Radians measure angles by arc length on the unit circle. A full circle is $2\pi$ radians (= $360^\circ$).

Degrees	Radians
$0^\circ$	$0$
$30^\circ$	$\pi/6$
$45^\circ$	$\pi/4$
$60^\circ$	$\pi/3$
$90^\circ$	$\pi/2$
$180^\circ$	$\pi$

To convert: $\text{radians} = \text{degrees} \cdot \dfrac{\pi}{180}$.

Key angles (memorize these)

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0$	$0$	$1$	$0$
$\pi/6$	$1/2$	$\sqrt{3}/2$	$1/\sqrt{3}$
$\pi/4$	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$\pi/3$	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$
$\pi/2$	$1$	$0$	undefined

Identities to know

Pythagorean identity — straight from the unit circle:

$\sin^2 \theta + \cos^2 \theta = 1$

Tangent in terms of sine and cosine:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

Double-angle:

$\sin(2\theta) = 2 \sin \theta \cos \theta$ $\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$

Signs by quadrant

Remember ASTC ("All Students Take Calculus"): in quadrant I All are positive; in II only Sine; in III only Tangent; in IV only Cosine.

Key takeaways

• $(\cos\theta, \sin\theta)$ are coordinates on the unit circle.
• $\sin^2\theta + \cos^2\theta = 1$ — the identity that solves a hundred problems.
• Memorize the table at $0, \pi/6, \pi/4, \pi/3, \pi/2$.