Question

Find exact values for sin θ and tan θ if cos θ = -4/9 and tan θ > 0.

Answer 1

Part 1)
`sin(\theta)=-(√(65))/(9)`

Part 2)
`tan(\theta)=(√(65))/(4)`

Explanation:

we have that

The cosine of angle theta is negative and the tangent of angle theta is positive

That means that the sine of angle theta is negative

step 1

Find
`sin(\theta)`

we know that

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

we have

`cos(\theta)=-(4)/(9)`

substitute

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

`sin^(2)(\theta) +(16)/(81)=1`

`sin^(2)(\theta)=1-(16)/(81)`

`sin^(2)(\theta)=(65)/(81)`

square root both sides

`sin(\theta)=\pm(√(65))/(9)`

Remember that

In this problem the sine of angle theta is negative

so

`sin(\theta)=-(√(65))/(9)`

step 2

Find
`tan(\theta)`

we know that

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

we have

`sin(\theta)=-(√(65))/(9)`

`cos(\theta)=-(4)/(9)`

substitute the given values

`tan(\theta)=-(√(65))/(9):-(4)/(9)=(√(65))/(4)`