Question

Given tan(theta)= -2 and pi/2 < theta < pi; find cos theta)

Answer 1

`\cos \theta=-(1)/(√(5))`

Explanation:

`Given\ \tan \theta=-2\ \ \ \ \ -(\pi)/(2)<\theta<\pi`

Hence
`\theta` is in second quadrant.

`\tan \theta=(opposite)/(adjacent)=-2\\\\Length\ of\ opposite=2,\ \ Length\ of\ adjacent=1\\`

Pythagorean Theorem:
`hypotenuse^2=opposite^2+adjacent^2`

`hypotenuse^2=(1)^2+(2)^2=1+4=5\\hypotenuse=√(5)`

`\cos \theta=(adjacent)/(hypotenuse)\\\\\cos \theta=(1)/(√(5))`

But in the second quadrant
`\cos \theta` will be negative.

Hence

`\cos \theta=-(1)/(√(5))`