Question

Given that `4<=y<=6`, find the value of y for which `2cos((2y)/3)+sqrt3=0`?

Answer 1

`color(blue)(y=(5pi)/4 , (7pi)/4)`

Explanation:

`2cos((2y)/3)+sqrt(3)=0`

`cos((2y)/3)=(-sqrt(3))/2`

Take arc cosine of both sides:

`arccos(cos((2y)/3))=arccos(-sqrt(3)/2)`

`(2y)/3=arccos(-sqrt(3)/2)`

`arccos(-sqrt(3)/2)=(5pi)/6 , (7pi)/6`

We try these two first, to see if we are in the given interval.

`(2y)/3=(5pi)/6=>y=(5pi)/4~~3.927`

`(2y)/3=(7pi)/6=>y=(7pi)/4~~5.498`

Any greater angles will not be in the given interval.

So:

`color(blue)(y=(5pi)/4 , (7pi)/4)`