cos and sin

[Giselleee3]

if cos (theta) = -1/2 and theta is in the third quadrant, what is the value of sin 2(theta) ?

 

[tkhunny]

\(\displaystyle \cos^{2}(\theta)+\sin^{2}(\theta)=1\)

 

[soroban]

Hello, Giselleee3!

\(\displaystyle \text{If }\cos\theta = \text{-}\tfrac{1}{2}\,\text{ and }\theta\text{ is in the third quadrant,}\)
\(\displaystyle \text{what is the value of }\sin 2\theta \:?}\)

\(\displaystyle \text{Since }\cos\theta = \text{-}\tfrac{1}{2},\text{ then: }\theta \:=\:\tfrac{2\pi}{3}\text{ or }\tfrac{4\pi}{3}\)
. . \(\displaystyle \theta\text{ is in Quadrant 3, so: }\theta \,=\,\tfrac{4\pi}{3}\)

\(\displaystyle \text{Then: }\,\sin2\theta \,=\,\sin\tfrac{8\pi}{3}\)

. . Can you finish?

 

[tkhunny]

Or is that \(\displaystyle \sin^{2}(\theta)\)?