Question

Using sum/difference trigonometry identies

Answer 1

π.

Explanation:

1) to simplify the given expression:

`2sin[0.5*( \theta +( \pi)/(6) + \theta-(7 \pi)/(6) )]*cos[0.5( \theta+( \pi)/(6) - \theta+(7 \pi)/(6))]=-1`

`2sin[ \theta-( \pi)/(2)]cos(2 \pi)/(3)=-1;\\sin[ \theta-( \pi)/(2)]=1;`

2) to resolve the simplified equation:

`\theta= \pi +2 \pi*n, \ n-Z.`

3) to calculate the number of the required solutions on [0; 2п):

when n=0 x=п - included in [0;2п)

when n=-1 x=-п - non-included in [0;2п)

4) finally, x=п.