The legislation of Cosines is provided to find the remaining parts of an slope (non-right) triangle as soon as either the lengths of two sides and also the measure up of the consisted of angle is recognized (SAS) or the lengths that the 3 sides (SSS) room known. In either of this cases, it is difficult to use the legislation of Sines since 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 other than for the third term and if C is a ideal angle the third term amounts to 0 because the cosine that 90 ° is 0 and also we get the Pythagorean Theorem. So, the Pythagorean organize is a special instance of the legislation of Cosines.

The law of Cosines can additionally be stated 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: 2 Sides and the consisted of Angle-SAS

offered a = 11 , b = 5 and m ∠ C = 20 ° . Find the continuing to be side and angles.


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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 uncover the remaining angles, that is easiest 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 the triangle is no a right triangle. So, when you take the train station you need to think about the obtuse angle who sine is 11 sin ( 20 ° ) 6.53 ≈ 0.5761 .




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instance 2: three Sides-SSS

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


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that is finest to find the angle opposite the longest next first. In this case, that is next b .

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

due to the fact that cos B is negative, we know that B is an obtuse angle. B ≈ 116.80 °

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

To discover the other two angles, that is most basic to usage the regulation of Sines.

a sin A = b sin B = c sin C

8 sin A ≈ 19 sin 116.80 ° ≈ 14 sin C