Suppose that cot θ = c and 0 < θ < π 2 . what is a formula for sin θ in terms of c?

cot θ  =  c ..............(1) [Given]

cot θ = 1 / tan θ.......(2)

Combining equations (1) and (2)

1 / tan θ  =  c

tan θ  =  1 / c

But tan θ  =  Opposite / Adjacent

Therefore,

opposite = 1, adjacent = c

Using the pythagora's theorem:

Hypotenuse²  =  Opposite²  +  Adjacent²

Hypotenuse²  =  1²  +  c²

sin θ = Opposite / Hypotenuse

Substituting for the opposite and hypotenuse in the sin θ formula above:

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Sinθ = 1/√c²+1 in terms of c

Step-by-step explanation:

Given cotθ = c and 0 < θ < 2π

Cot θ = 1/tanθ = c

tanθ = 1/c

From SOH, CAH, TOA

tanθ = opposite/adjacent = 1/c

This shows that opposite = 1

adjacent = c

Sinθ = opposite/hypothenuse

Before we can get sinθ, we need the hypotenuse side. Using Pythagoras theorem to get the hypotenuse. According to the theorem, adj²+opp² = hyp²

hyp² = c²+1²

hyp² = c²+1

hyp = √c²+1

Sinθ = 1/√c²+1 in terms of c

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