Basic identities

Identities are true for every angle. The main one is just the Pythagorean theorem on the unit circle: $x^{2} + y^{2} = 1$ , so $cos^{2} θ + sin^{2} θ = 1$ .

Identity	Use
$sin^{2} θ + cos^{2} θ = 1$	Find one from the other
$tan θ = \frac{sin θ}{cos θ}$	Rewrite tangent
$sin (- θ) = - sin θ$	Sine is odd
$cos (- θ) = cos θ$	Cosine is even
$sin 2 θ = 2 sin θ cos θ$	Double angle

Work through it

Simplify $\frac{1 - cos ^{2} θ}{sin θ}$ .

Replace the top. $\frac{sin ^{2} θ}{sin θ}$

Cancel. $sin θ$

$cos θ = \frac{5}{13}$ in Quadrant I. Find $sin 2 θ$ .

Find sine. $sin θ = \frac{12}{13}$

Double angle. $2 \cdot \frac{12}{13} \cdot \frac{5}{13}$

Multiply. $\frac{120}{169}$

Watch out for this

$sin 2 θ = 2 sin θ$ .

Check $θ = 9 0^{\circ}$ : $sin 18 0^{\circ} = 0$ , but $2 sin 9 0^{\circ} = 2$ .

The short version

sin^{2} θ + cos^{2} θ = 1

and

tan θ = \frac{sin θ}{cos θ}

. Rewrite things in sine and cosine when you get stuck.

Your turn

PreviousExact values on the unit circle NextUnit check

Want help with this?

Book a free consultation, and we'll help you make a plan.

Book a free consultation