If sinθ = 4/5, what is the value of cot θ + cosec θ?

Question

If sinθ = \(\frac{4}{5}\), what is the value of cot θ + cosec θ?

Answer 1

Given: sinθ = \(\frac{4}{5}\)

To find: cot θ + cosec θ 

∵ sin²θ + cos²θ = 1 

∴ cos²θ = 1 – sin²θ

⇒ cosθ = \(\sqrt{1-sin^2 θ}\)

⇒ cosθ = \(\sqrt{1-\Big(\frac{4}{5}\Big)^2}\)   = \(\sqrt{1-\frac{16}{25}}\) 

=  \(\sqrt{1-\frac{25-16}{25}}\) = \(\sqrt{\frac{9}{25}}=\frac{3}{5}\) 

Now, as cotθ = \(\frac{cosθ}{sinθ} = \frac{3/5}{4/5}=\frac{3}{4}\) 

Also, cosecθ = \(\frac{1}{sinθ}=\frac{1}{4/5}=\frac{5}{4}\) 

cotθ + cosecθ = \(\frac{3}{4}+\frac{5}{4}=\frac{3+5}{4}\) = \(\frac{8}{4}\) = 2