Law of sines and cosines

SOH-CAH-TOA needs a right angle. These two laws work on every triangle.

Side

a

faces angle

A

, side

b

faces

B

and side

c

faces

C

Law of sines:

\frac{a}{sin A} = \frac{b}{sin B}

. Law of cosines:

c^{2} = a^{2} + b^{2} - 2 ab cos C

Work through it

Set up sines. $\frac{4}{sin 3 0 ^{\circ}} = \frac{b}{sin 9 0 ^{\circ}}$

Use $sin 3 0^{\circ} = \frac{1}{2}$ and $sin 9 0^{\circ} = 1$ . $8 = b$

Use cosines. $c^{2} = 9 + 64 - 2 (3) (8) cos 6 0^{\circ}$

Use $cos 6 0^{\circ} = \frac{1}{2}$ . $c^{2} = 73 - 24 = 49$

Take the root. $c = 7$

Pairing a side with the wrong angle in the law of sines.

Each fraction uses a side and the angle facing it.

The short version
Have a side and its opposite angle? Use sines. Have two sides and the angle between, or all three sides? Use cosines.