The law of Cosines is provided to uncover the remaining parts of an tilt (non-right) triangle when either the lengths of 2 sides and the measure of the consisted of angle is recognized (SAS) or the lengths that the 3 sides (SSS) space known. In either of these cases, the is difficult to usage the legislation of Sines because we cannot set up a solvable proportion.

The legislation of Cosines states:

c 2 = a 2 + b 2 − 2 a b     cos C .

This resembles the Pythagorean theorem except for the 3rd term and if C is a right angle the third term equals 0 since the cosine the 90 ° is 0 and we gain the Pythagorean Theorem. So, the Pythagorean theorem is a special situation of the regulation of Cosines.

The legislation 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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example 1: two Sides and also the had Angle-SAS

given a = 11 , b = 5 and m ∠ C = 20 ° . Discover the remaining side and also 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 staying angles, that is easiest to now 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 °

keep in mind that angle A is opposite to the longest side and also the triangle is not a appropriate triangle. So, when you take the train station you need to consider the obtuse angle whose sine is 11 sin ( 20 ° ) 6.53 ≈ 0.5761 .




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instance 2: 3 Sides-SSS

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


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

because cos B is negative, we know that B is an obtuse angle. B ≈ 116.80 °

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

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

a sin A = b sin B = c sin C

8 sin A ≈ 19 sin 116.80 ° ≈ 14 sin C