Law of Sines and Cosines

How to determine which formula to use

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

Law Of sines case I
Law Of sines case II
Law Of cosines case I
Law Of cosines case II

Practice Problems

Problem 1

Can you use the the law of sines, the law of cosines , or neither to solve the unknown side triangle 1?

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})} $

Problem 2

Can you use the the law of sines, the law of cosines , or neither to solve the unknown side triangle 1?

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 ) $

Problem 3

Can you use the the law of sines, the law of cosines , or neither to solve the unknown side triangle 1?

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} $

Problem 4

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

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))

Problem 5

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

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))

Problem 6

Decide which formula (law of Sines/Cosines) you would use to calculate the value of x below? After you decide that, try to set up the equation(Do not solve--just substitute into the proper formula)

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


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