Question

Use identities to find tan, csc, sec, and cot. Where necessary, rationalize denominators

Answer 1

We are given:

`sin\text{ }\theta=(3)/(5),cos\text{ }\theta=(4)/(5)`

The tangent is defined as the ratio of the sine and the cosine:

`tan\text{ }\theta=\frac{sin\text{ }\theta}{cos\text{ }\theta}`

Calculating:

The cotangent is the reciprocal of the tangent:

`\begin{gathered} \cot\theta=(1)/(\tan\theta) \\ \\ \cot\theta=(1)/((3)/(4))=(4)/(3) \end{gathered}`

The secant is the reciprocal of the cosine:

`\begin{gathered} \sec\theta=(1)/(\cos\theta)=(1)/((4)/(5)) \\ \\ \sec\theta=(5)/(4) \end{gathered}`

The cosecant is the reciprocal of the sine:

`\begin{gathered} \csc\theta=(1)/(\sin\theta)=(1)/((3)/(5)) \\ \\ \csc\theta=(5)/(3) \end{gathered}`