Question

If tan(theta) = -1/2 and pi/2 < theta < pi, what is cos(theta)?

Answer 1

`\cos \theta=-(2√(5))/(5)`

Explanation:

It was given that:

`\tan(\theta)=-(1)/(2)`

and
`(\pi)/(2)\le \theta \le \pi`.

We use the relation:

`\sec^2 \theta=1+\tan^2\theta`

We substitute the value to get:

`\sec^2 \theta=1+((1)/(2))^2`

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

`\sec^2 \theta=(5)/(4)`

`\sec \theta=\pm (√(5))/(2)`

Reciprocate both sides to get

`\cos \theta=\pm (2)/(√(5))`

Rationalize the denominator:

`\cos \theta=\pm (2√(5))/(5)`

The given interval,
`(\pi)/(2)\le \theta \le \pi` is the same as the 2nd quadrant where the cosine ratio is negative.

`\therefore \cos \theta=-(2√(5))/(5)`