Question

Find exact values for sin θ, cos θ and tan θ if csc θ = 3/2 and cos θ < 0.

Answer 1

Part 1)
`sin(\theta)=(2)/(3)`

Par 2)
`cos(\theta)=-(√(5))/(3)`

Part 3)
`tan(\theta)=-(2√(5))/(5)`

Explanation:

step 1

Find the
`sin(\theta)`

we have

`csc(\theta)=(3)/(2)`

Remember that

`csc(\theta)=(1)/(sin(\theta))`

therefore

`sin(\theta)=(2)/(3)`

step 2

Find the
`cos(\theta)`

we know that

`sin^(2)(\theta) +cos^(2)(\theta)=1`

we have

`sin(\theta)=(2)/(3)`

substitute

`((2)/(3))^(2) +cos^(2)(\theta)=1`

`(4)/(9) +cos^(2)(\theta)=1`

`cos^(2)(\theta)=1-(4)/(9)`

`cos^(2)(\theta)=(5)/(9)`

square root both sides

`cos(\theta)=\pm(√(5))/(3)`

we have that

`cos(\theta) < 0` ---> given problem

so

`cos(\theta)=-(√(5))/(3)`

step 3

Find the
`tan(\theta)`

we know that

`tan(\theta)=(sin(\theta))/(cos(\theta))`

we have

`sin(\theta)=(2)/(3)`

`cos(\theta)=-(√(5))/(3)`

substitute

`tan(\theta)=(2)/(3):-(√(5))/(3)=-(2)/(√(5))`

Simplify

`tan(\theta)=-(2√(5))/(5)`