Question

The line with equation a + 4b = 0 coincides with the terminal side of an angle θ in standard position and cos θ>0 . What is the value of sinθ

Answer 1

`a+4b=0\implies a=-4b`

Treat
`a` as a function of
`b`, so that any point on this line takes the form
`(b,-4b)`. Suppose
`b` is positive; then any such point lies in the 4th quadrant, and this guarantees that the angle
`\theta` has a positive value for
`\cos\theta`.

By definition of tangent and cotangent, we have

`\tan\theta=\frac{-4b}b=-4\implies\cot\theta=-\frac14`

Recall the Pythagorean identity,

`\cot^2\theta+1=\csc^2\theta`

In the 4th quadrant, we have
`\sin\theta<0`, so that
`\csc\theta<0` as well. So when we solve for
`\csc\theta` above, we need to take the negative square root:

`\csc\theta=-√(\cot^2\theta+1)=-\frac{√(17)}4`

`\implies\sin\theta=-\frac4{√(17)}`

Answer 2

`-(√(17))/(17)`

Explanation:

got it wrong on the test but luckily it shows me which one is right ;P