Sine and Cosine Rule Trig Identities

The sine rule gives the relation between the angles and the corresponding sides of a triangle. For the non-right-angled triangles, we will have to use the sine rule and the cosine rule. For a triangle with sides ‘a’, ‘b’, and ‘c’ and the respective opposite angles are A, B, and C, sine rule can be given as,

• a/sinA = b/sinB = c/sinC
• sinA/a = sinB/b = sinC/c
• a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

The cosine rule gives the relation between the angles and the sides of a triangle and is usually used when two sides and the included angle of a triangle are given. Cosine rule for a triangle with sides ‘a’, ‘b’, and ‘c’ and the respective opposite angles are A, B, and C, sine rule can be given as,

• a² = b² + c² – 2bc·cosA
• b² = c² + a² – 2ca·cosB
• c² = a² + b² – 2ab·cosC

Important Notes on Trigonometric Identities

• To write the trigonometric ratios of complementary angles, we consider the following as pairs: (sin, cos), (cosec, sec), and (tan, cot).
• While writing the trigonometric ratios of supplementary angles, the trigonometric ratio won’t change. The sign can be decided using the fact that only sin and cosec are positive in the second quadrant where the angle is of the form (180-θ).
• There are 3 formulas for the cos 2x formula. Among them, you can remember just the first one because the other two can be obtained by the Pythagorean identity sin²x + cos²x = 1.
• The half-angle formula of tan is obtained by applying the identity tan = sin/cos and then using the half-angle formulas of sin and cos.