7.3 The Law of Cosines

Law of Cosines Equations

If C = 90° in the third form of the law of cosines, then cos C = 0, and the formula becomes c² = a² + b², the Pythagorean theorem

Triangle Side Length Restriction

In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side

THREE: Given the triangle with the following: 1st side = 30 2nd side = 25. side a 3rd side = 18 Angle opposite side measuring 25 = A Find A (round answer to 4 decimal places)

This is Case 4, a SSS, because all sides are known. Find the measure of angle A using the law of cosine for a

ONE: Find c if a = 2.25 mi, b = 3.93 mi and ∠ C = 42° c =

This is Case 3 because it's a SAS, two sides and the included angle are known. So find the missing side c using the law of cosines. *we only take the positive square root because we're finding the length of a side.

FIVE: Given the triangle with the following: 1st side = 21 2nd side = 28 3rd side = x Angle opposite side of x = 45° Find x (round answer to nearest tenth)

This is Case 3 because it's a SAS, two sides and the included angle are known. So find the missing side using the law of cosines. *we only take the positive square root because we're finding the length of a side.

FOUR: Given the triangle with the following: 1st side = 24 2nd side = 23 3rd side = x Angle opposite side of x = 28° Find x (round answer to 3 decimal places)

This is Case 3 because it's a SAS, two sides and the included angle are known. So find the missing side using the law of cosines. *we only take the positive square root because we're finding the length of a side.

TWO: Given the triangle with the following: 1st side = 21 2nd side = 19 3rd side = x Angle opposite side of x = 22° Find x (round answer to 4 decimal places)

This is Case 3 because it's a SAS, two sides and the included angle are known. So find the missing side using the law of cosines. *we only take the positive square root because we're finding the length of a side.

Example 2 (1 of 2) Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m

Caution: Had we used the law of sines to find C rather than B in Example 2, we would not have known whether C was equal to 81.7° or its supplement, 98.3.

SIX: Solve the triangle (thus find all angles and side measures) and round answers to the nearest thousandth 1st side (side c) = 16 2nd side = unknown 3rd side = unknown Angle B = 47° Angle C (opposite side c) = 90 Angle A = unknown Find x (round answer to 4 decimal places)

This is Case 1 because it's a SAA, one side and the two angles are known. To find all sides and angles: 1. Find the third angle using the angle sum formula (bc angle C is 90 degrees): A + B + C = 180° 2. Find the remaining sides using the law of sines. *we only take the positive square root because we're finding the length of a side.