Question

Given sin theta = 2/3 where pi/2 < or = theta < or = pi find the value of sin 2 theta, please help me

Answer 1

First, you need to find \(\cos(\theta)\). Fortunately, since we know that \(\cos^2(\theta)+\sin^2(\theta)=1\) and we know \(\sin(\theta)\). We can find \(\cos(\theta)\)

Answer 2

\[\cos^2(\theta)=1-\left({2 \over 3}\right)^2=1-{4 \over 9}={5 \over 9}\]Thus, \[\cos(\theta)=\sqrt{5 \over 9}\]

Answer 3

Additionally, since we have \(\pi/2 \leq \theta \leq \pi\), we know that we must take the negative of this square root. So\[\cos(\theta)=-\sqrt{5 \over 9}\]

Answer 4

Finally use the formula \[\sin(2\theta)=2\sin(\theta)\cos(\theta)\]To find finish the problem.