Compound Angle Formulas

[MathDefinition title="Compound Angle Formulas"] Compound angle formulas express trigonometric functions of sum or difference of two angles in terms of trigonometric functions of individual angles. [/MathDefinition]

[MathKeyFormula title="Sum and Difference Formulas"]

1. sin(A + B) = sin A cos B + cos A sin B
2. sin(A − B) = sin A cos B − cos A sin B
3. cos(A + B) = cos A cos B − sin A sin B
4. cos(A − B) = cos A cos B + sin A sin B
5. tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
6. tan(A − B) = (tan A − tan B) / (1 + tan A tan B) [/MathKeyFormula]

[MathNote title="Memory Trick"] For sin(A ± B): sine starts, signs match (sin cos ± cos sin) For cos(A ± B): cosine starts, signs opposite (cos cos ∓ sin sin) [/MathNote]

[MathExample title="Example 1: Finding sin 75°"] Problem: Find the value of sin 75°.

Solution: sin 75° = sin(45° + 30°)

Using sin(A + B) = sin A cos B + cos A sin B: sin 75° = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = √3/(2√2) + 1/(2√2) = (√3 + 1)/(2√2) = (√3 + 1)/(2√2) × (√2/√2) = (√6 + √2)/4

Therefore, sin 75° = (√6 + √2)/4 [/MathExample]

[MathExample title="Example 2: Finding cos 15°"] Problem: Find the value of cos 15°.

Solution: cos 15° = cos(45° − 30°)

Using cos(A − B) = cos A cos B + sin A sin B: cos 15° = cos 45° cos 30° + sin 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = √3/(2√2) + 1/(2√2) = (√3 + 1)/(2√2) = (√6 + √2)/4

Therefore, cos 15° = (√6 + √2)/4 [/MathExample]

[MathExample title="Example 3: Proving Identity"] Problem: Prove that sin(A + B) / sin(A − B) = (tan A + tan B) / (tan A − tan B).

Solution: LHS = sin(A + B) / sin(A − B) = (sin A cos B + cos A sin B) / (sin A cos B − cos A sin B)

Dividing numerator and denominator by cos A cos B: = (sin A cos B/(cos A cos B) + cos A sin B/(cos A cos B)) / (sin A cos B/(cos A cos B) − cos A sin B/(cos A cos B)) = (tan A + tan B) / (tan A − tan B) = RHS

Hence proved. [/MathExample]

[MathExample title="Example 4: Finding tan 105°"] Problem: Find the value of tan 105°.

Solution: tan 105° = tan(60° + 45°)

Using tan(A + B) = (tan A + tan B) / (1 − tan A tan B): tan 105° = (tan 60° + tan 45°) / (1 − tan 60° tan 45°) = (√3 + 1) / (1 − √3 × 1) = (√3 + 1) / (1 − √3)

Rationalizing the denominator: = (√3 + 1)(1 + √3) / ((1 − √3)(1 + √3)) = (√3 + 3 + 1 + √3) / (1 − 3) = (4 + 2√3) / (−2) = −2 − √3

Therefore, tan 105° = −2 − √3 [/MathExample]