Using the Law of Sines and Cosines to Find Angles – Practice ProblemsHaving fun while studying, practice your skills by solving these exercises!

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For any non-right triangle, if two sides together with the angle opposite to one of them are known, then you can use the law of sines to compute the angle opposite to the other known side.

For a given triangle with side lengths a, b, and c, the law of sines states that

a b c
--------- = --------- = ---------,
sin A sin B sin C

where A, B, and C are the angles opposite to a, b, and c, respectively. You just need to make use of the relevant pair of ratios and remember that, when working with proportions, the product of the means equals the product of the extremes.

If only all three sides lengths are given, and you need to look for an angle, then you can use the law of cosines.

For a given triangle with side lengths a, b, and c, the law of cosines states that

c2 = a2 + b2 – 2ab(cosC)
b2 = a2 + c2 – 2ac(cosB)
a2 = b2 + c2 – 2bc(cosA)

where A, B, and C are the angles opposite to a, b, and c, respectively.

Apply trigonometry to general triangles.

CCSS.MATH.CONTENT.HSG.SRT.D.11

Exercises in this Practice Problem
 Calculate the angle using the law of sines. Find angle $A$ using the law of cosines. Determine the law of cosines equation. Solve for the angle Pete needs to navigate to the island. Define the law of cosines. Identify the values of the angles marking potential new locations for lighthouses.