Trigonometric Identities

• Sin nπ = 0 if n∈I
• Cos nπ/2 = 0 if n is an odd integer
• cos (nπ + θ) = (-1)n cos θ, n ∈ I
• sin (nπ + θ) = (-1)n sin θ, n ∈ I
• Cos (nπ/2 +θ) = (−1)^{(n + 1)/2} Sinθ, if n is odd integer
• Sin (nπ/2 +θ) = (−1)^(n-1/2) Cosθ, if n is odd integer

For Any Three Angles A,B,C

(a) sin (A+B+C) = sin A cos B cos C + sin B cos C cos A + sin C cos A cos B − sin A sin B sin C

(b) cos (A+B+C) = cos A cos B cos C – cos A sin B sin C - cos B sin A sin C – cos C sin A sin B

(c) tan (A+B+C) =

(d) cot (A+B+C) =

Greatest And Least Values of a cosθ + b sinθ

Some Important Identities

If A+B+C = π then

1. sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
2. cos 2A + cos 2B + cos 2C = -1 –4 cos A cos B cos C
3. sin A + sin B + sin C = 4 CosA/2 cos B/2 cosC/2
4. cos A + cos B + cos C = 1 + 4 sin A/2 sin B/2 sin C/2
5. tan A + tan B + tan C = tan A tan B tan C
6. cot A cot B + cot B cot C + cot C cot A = 1
7. tan A/2 tan B/2 + tan B/2 tan C/2 + tan C/2 tan A/2 = 1
8. cot A/2 + cot B/2 + cot C/2 = cot A/2. cot B/2. cot C/2

Two Useful Identities

1. cos α + cos β + cos γ + cos (α+β+γ) = 4
2. sin α + sin β + sin γ - sin (α+β+γ) = 4