The legislation of Cosines is provided to discover the remaining components of an oblique (non-right) triangle once either the lengths of 2 sides and also the measure of the contained angle is known (SAS) or the lengths that the three sides (SSS) are known. In one of two people of these cases, it is impossible to use the law of Sines due to the fact that we cannot set up a solvable proportion.

The legislation of Cosines states:

c 2 = a 2 + b 2 − 2 a b     cos C .

This each other the Pythagorean to organize except for the third term and if C is a ideal angle the third term equals 0 since the cosine of 90 ° is 0 and also we obtain the Pythagorean Theorem. So, the Pythagorean to organize is a special case of the law of Cosines.

The regulation of Cosines can additionally be proclaimed as

b 2 = a 2 + c 2 − 2 a c     cos B or

a 2 = b 2 + c 2 − 2 b c     cos A .

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instance 1: two Sides and the had Angle-SAS

offered a = 11 , b = 5 and also m ∠ C = 20 ° . Uncover the staying side and also angles.

c 2 = a 2 + b 2 − 2 a b     cos C

c = a 2 + b 2 − 2 a b     cos C

= 11 2 + 5 2 − 2 ( 11 ) ( 5 ) ( cos 20 ° )

≈ 6.53

To discover the staying angles, that is simplest to currently use the regulation of Sines. sin A ≈ 11 sin 20 ° 6.53

A ≈ 144.82 °

sin B ≈ 5 sin 20 ° 6.53

B ≈ 15.2 °

note that edge A is opposite come the longest side and also the triangle is not a right triangle. So, as soon as you take it the station you require to take into consideration the obtuse angle whose sine is 11 sin ( 20 ° ) 6.53 ≈ 0.5761 .

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example 2: 3 Sides-SSS

provided a = 8 , b = 19 and c = 14 . Discover the actions of the angles.

the is ideal to find the angle opposite the longest side first. In this case, the is side b .

cos B = b 2 − a 2 − c 2 − 2 a c = 19 2 − 8 2 − 14 2 − 2 ( 8 ) ( 14 ) ≈ − 0.45089

since cos B is negative, we know that B is one obtuse angle. B ≈ 116.80 °

since B is one obtuse angle and also a triangle has at most one obtuse angle, we know that angle A and also angle C space both acute.

To discover the various other two angles, that is easiest to usage the law of Sines.

a sin A = b sin B = c sin C

8 sin A ≈ 19 sin 116.80 ° ≈ 14 sin C