The goal of this page is to help students better understand when to use the law of sines and when to use the Law of Cosines

#### When to Use

**Practice Problems**

Law of Sines

Remember, the law of sines is all about opposite pairs.

In this case, we have a side of length 11 opposite a known angle of $$ 29^{\circ} $$ (first opposite pair) and we want to find the side opposite the known angle of $$ 118^\circ$$.

$ \frac{\red x} {sin(118^{\circ})} = \frac{11}{ sin(29^{\circ})} $

Law of Cosines Remember, the law of cosines is all about included angle ( or knowing 3 sides and wanting to find an angle).

In this case, we have a side of length 20 and of 13 and the included angle of $$ 66^\circ$$.

$ \red a^2 = b^2 + c^2 - 2bc \cdot cos( \angle a ) \\ \red a^2 = 20^2 + 13^2 - 2\cdot 20 \cdot 13 \cdot cos( 66 ) $

Law of Sines

Remember, the law of sines is all about opposite pairs.

In this case, we have a side of length 16 opposite a known angle of $$ 115^{\circ} $$ (first opposite pair) and we want to find the angle opposite the known side of length 32. We can set up the proportion below and solve :

$ \frac{sin(115^{\circ})}{16} = \frac{sin(\red x)}{32} $

Since you know 3 sides, and are trying to find an angle this is Law of Cosines problem. 8² = 5² + 6² -2(5)(6)(cos(x))

Since you know 2 sides , their included angle, and you are trying to find the side length opposite the angle, this is Law of Cosines problem. x² = 11² + 7² -2(11)(7)(cos(50))

Since you know a side length (11) and its opposite angle (50) and want to the angle measurement opposite the length of side 7, this is a law of sines problem