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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