Cosine rule to find a side

c² = a² + b² − 2ab cos C for the side opposite angle C.

Key formulas

You need the angle between the two known sides. That angle is opposite the side you are finding.

a=5, b=7, C=60°. c²=25+49−2×5×7×½=74−35=39 → c=√39.

If C>90°, cos C is negative, so −2ab cos C becomes positive and c is longer — as expected.

Problem. In △ABC, a = 5, b = 7 and included angle C = 60°. Find side c.

Solution. c² = a² + b² − 2ab cos C = 25 + 49 − 2·5·7·(1/2) = 74 − 35 = 39. So c = √39.

If C=90°, cos0 term vanishes and you recover Pythagoras.

a=4, b=5, C=90°. Then c equals:

Yes — set the calculator to degrees unless the angle is clearly in radians.