Answer:
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
[tex]sin \theta = \frac{1}{\sqrt{1 + c^2} }[/tex]
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
[tex]Hypotenuse = \sqrt{1 + c^2}[/tex]
sin θ = Opposite / Hypotenuse
Substituting for the opposite and hypotenuse in the sin θ formula above:
[tex]sin \theta = \frac{1}{\sqrt{1 + c^2} }[/tex]
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