Question

Would someone mind explaining to me why I got this question wrong? Thank you so much for your help!

Answer 1

`-sqrt5/5`

Explanation:

`"using the "color(blue)"half angle identity for cos"`

`•color(white)(x)cos(theta/2)=+-sqrt((1+costheta)/2)`

`"given "pi< theta <(3pi)/2" then"`

`pi/2< theta/2< (3pi)/4larrcolor(blue)"second quadrant"`

`"where "cos(theta/2)<0`

`cos(theta/2)=-sqrt((1-3/5)/2)`

`color(white)(xxxxxx)=-sqrt(1/5)`

`color(white)(xxxxxx)=-1/sqrt5xxsqrt5/sqrt5=-sqrt5/5`

Answer 2

See explanation.

Explanation:

Think about it logically. `pi < theta < (3pi)/2`, so `theta` lies in the 3rd quadrant. To get to the 3rd quadrant, you take the basic acute angle and add `pi` or `180^@` to it. If you halve an angle between `pi` and `(3pi)/2`, it will always be greater than `pi/2` because the original angle was greater than `pi` to begin with. Now, realize that since `theta/2` is in the 2nd quadrant, `cos(theta/2)` will always be negative because `costheta` is negative in the 2nd and 3rd quadrant. This is why the correct answer should be `-sqrt5/5`.