Law of Sines vs. Law of Cosines: Which One Should You Use?

In triangle $A B C$ , side $a$ is opposite angle $A$ , side $b$ is opposite angle $B$ , and side $c$ is opposite angle $C$ .

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

a^{2} = b^{2} + c^{2} - 2 b c cos A

How to choose

Have an angle and its opposite side? Use the Law of Sines.
Have two sides and the angle between them? Use the Law of Cosines.
Have all three sides? Use the Law of Cosines to find an angle.

Law of Sines: $A = 3 0^{\circ}$ , $a = 10$ , $B = 4 5^{\circ}$ . Find $b$ .

$\frac{b}{s i n 4 5 ^{\circ}} = \frac{10}{s i n 3 0 ^{\circ}}$

$b = \frac{10 s i n 4 5 ^{\circ}}{s i n 3 0 ^{\circ}} \approx 14.14$

Law of Cosines: $b = 5$ , $c = 7$ , $A = 6 0^{\circ}$ . Find $a$ .

$a^{2} = 5^{2} + 7^{2} - 2 (5) (7) cos 6 0^{\circ} = 39$

$a = 39 \approx 6.24$

Three sides: $a = 7$ , $b = 8$ , $c = 9$ . Find angle $C$ .

$9^{2} = 7^{2} + 8^{2} - 2 (7) (8) cos C$

$cos C = \frac{2}{7}$ , so $C \approx 73. 4^{\circ}$

Common mistake

Using the Law of Sines to find an angle and stopping at the first answer.

When you know two sides and a non-included angle, there may be two possible triangles because $sin θ = sin (18 0^{\circ} - θ)$ . Check whether the second angle also works.

Try it yourself

$A = 4 0^{\circ}$ , $a = 12$ , $B = 6 5^{\circ}$ . Find $b$ .
Show answer
$b \approx 16.92$
$b = 6$ , $c = 9$ , $A = 5 0^{\circ}$ . Find $a$ .
Show answer
$a \approx 6.90$
$a = 5$ , $b = 6$ , $c = 7$ . Find angle $C$ .
Show answer
$C \approx 78. 5^{\circ}$