Question

How do you do this trig problem.

Answer 1

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

`\sin\theta=\frac35\implies\theta=\sin^(-1)\frac35`

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=\frac45`

because

`\sin^2\theta+\cos^2\theta=\frac9{25}+(16)/(25)=1`

Then in this triangle,

`\cot\theta=\cot\left(\sin^(-1)\frac35\right)=(\cos\theta)/(\sin\theta)=(\frac45)/(\frac35)=\frac43`