Question

How do you do this trig problem.

Answer 1

Consider a right triangle in which one of the angles $\theta$ satisfies

$\sin\theta=\dfrac35\implies\theta=\sin^{-1}\dfrac35$

That is, the side opposite $\theta$ occurs in a ratio of 3 to 5 with the hypotenuse. The side adjacent to $\theta$ then occurs in a ratio of 4 to 5 with the hypotenuse. In other words,

$\cos\theta=\dfrac45$

because

$\sin^2\theta+\cos^2\theta=\dfrac9{25}+\dfrac{16}{25}=1$

Then in this triangle,

$\cot\theta=\cot\left(\sin^{-1}\dfrac35\right)=\dfrac{\cos\theta}{\sin\theta}=\dfrac{\frac45}{\frac35}=\dfrac43$