If cos9a = sina, Find tan5a — Trigonometry Problem

Given: cos9a = sina

Step 1: Use the complementary angle identity We know: sin θ = cos(90° − θ)

So: sina = cos(90° − a)

Step 2: Substitute in the equation cos9a = cos(90° − a)

Step 3: Equate the angles Since cos is a one-to-one function for acute angles: 9a = 90° − a

Step 4: Solve for a 9a + a = 90° 10a = 90° a = 9°

Step 5: Find tan5a tan5a = tan(5 × 9°) = tan45°

tan45° = 1

Answer: tan5a = 1

Verification: • cos9a = cos(9 × 9°) = cos81° • sina = sin9° • cos81° = sin9° ✓ (since cos81° = cos(90°−9°) = sin9°)

Complementary angle identities (used in this problem): • sin θ = cos(90° − θ) • cos θ = sin(90° − θ) • tan θ = cot(90° − θ) • cot θ = tan(90° − θ) • sec θ = cosec(90° − θ) • cosec θ = sec(90° − θ)

These identities are called co-function identities because the functions of complementary angles are equal.

Special angle values: • tan45° = 1 • sin45° = cos45° = 1/√2 • tan30° = 1/√3; tan60° = √3 • tan0° = 0; tan90° = undefined

Similar problem types: • If sin3a = cos7a → sin3a = sin(90−7a) → 3a = 90−7a → 10a = 90 → a = 9° • If tan2a = cot4a → tan2a = tan(90−4a) → 2a = 90−4a → 6a = 90 → a = 15°