1. If sin θ = 5/13 . where θ is an acute angle, find the value of other trigonometric ratios using identities.

Solution: sinθ = 5/13 [Given]

∴  cosecθ = 1/sinθ = 1÷ (5/13)

∴ cosec θ = 13/5

We know that,

sin²θ + cos²θ = 1

∴  (5/13)² + cos²θ = 1

∴  25/169 + cos²θ = 1

∴ cos²θ= 1 – 25/169

∴  cos²θ = (169 – 25 )/169

∴  cos²θ= 144/169

∴ cosθ = ±√ (144/169)

∴ cosθ = ±12/13

But, θ is an acute angle [Given]

∴ All trigonometric ratios must be positive,

∴ cosθ = 12/13

secθ = 1/cosθ

∴ sec θ = 1÷ (12/13)

∴ secθ = 13/12

tanθ = sinθ ÷ cosθ

∴ tanθ = 5/13 ÷ 12/13

∴ tan θ = 5/13 × 13/12.

∴ tanθ = 5/12

cotθ = 1/tanθ

∴ cot θ = 1÷ (5/12)

∴ cot θ = 12/5