The restrictions mean that thetaθ is in the second quadrant.
Hence, sine will be positive and cosine will be negative.
The expansion of sin2thetasin2θ will be 2sinthetacostheta2sinθcosθ, following the double angle identity. So, we need to determine the value of sinthetasinθ.
Recall that costheta= "adjacent"/"hypotenuse"cosθ=adjacenthypotenuse, so "adjacent" = -24adjacent=−24 and "hypotenuse" = 25hypotenuse=25. By pythagorean theorem, we have that:
o^2 + (-24)^2 = 25^2o2+(−24)2=252, where the side opposite theta = oθ=o
o^2 = 625 - 576o2=625−576
o^2 = 49o2=49
o = +-7o=±7
However, we know that in quadrant 2, sine is positive, so the positive answer is the correct one.
Now, recall that sintheta = "opposite"/"hypotenuse" = 7/25sinθ=oppositehypotenuse=725
Therefore, we can calculate sin2thetasin2θ as the following:
sin2theta = 2sinthetacostheta = 2(7/25)(-24/25) = -336/25sin2θ=2sinθcosθ=2(725)(−2425)=−33625
Hopefully this helps!
