Question

Please help me with this question

Answer 1

`\displaystyle{ \theta = (5\pi)/(6) ,(7\pi)/(6)}`

Explanation:

Given the equation:

`\displaystyle{2 \cos \theta = - √(3) }`

Divide both sides by 2:

`\displaystyle{ \cos \theta = \frac{- √(3)} {2}}`

Cosine is negative in Quadrant 2 and 3. Therefore, we have to find measurement that satisfies the equation. We know that:

`\displaystyle{ \cos (\pi)/(6) = ( √(3) )/(2)}`

π/6 is a reference angle. Therefore, to find √3/2 in negative form (for cosine), we have to subtract π by π/6 (Q2) and also add π by π/6 (Q3) as well.

Q2

`\displaystyle{ \pi - (\pi)/(6) = (6\pi)/(6) - (\pi)/(6)} \\ \\ \displaystyle{ \pi - (\pi)/(6)= (5\pi)/(6)}`

Q3

`\displaystyle{\pi + (\pi)/(6) = (6\pi)/(6) + (\pi)/(6)} \\ \\ \displaystyle{\pi + (\pi)/(6) = (7\pi)/(6)}`

Since the interval is from 0 to 2π, only two measurements above (5π/6 and 7π/6) are only solutions to the equation.

`\displaystyle{ \therefore \theta = (5\pi)/(6) ,(7\pi)/(6)}`