1. ## Exact values of...

Find the exact values of $sin2u$ and $cos \frac{u}{2}$.

a. $cscu = 3, restriction: \frac{\pi}{2} < u < \pi$

b. $tanu = -\frac{8}{5}, restriction: \frac{3\pi}2 < u < 2\pi$

Please solve it step by step so I can understand it too. xD

2. Originally Posted by Silvahere
Find the exact values of $sin2u$ and $cos \frac{u}{2}$.

a. $cscu = 3, restriction: \frac{\pi}{2} < u < \pi$
"cosecant" is "hypotenuse over opposite side" so imagine "hypotenuse" of length 3 and "opposite side" of length 1. You can calculate the length of the "near side". Since it is in the second quadrant,that number should be negative. Now just find cos(u) and sin(u) from that triangle and use the trig identities: sin(2u)= 2sin(u)cos(u) and $cos(u/2)= \pm\sqrt{1+ cos(u)}$. You can get the sign from the fact that if u is in the second quadrant, u/2 is in the first quadrant.

b. $tanu = -\frac{8}{5}, restriction: \frac{3\pi}2 < u < 2\pi$
tangent is "opposite side over near side" so imagine a triangle with "opposite side" of length 8 and "near side" of length 5. You can use the Pythagorean theorem to get the length of the hypotenuse and so sin(u) and cos(u). Since u is in the fourth quadrant, sine is negative and cosine is positive. Since u is in the fourth quadrant, u/2 is in the second quadrant, where cosine is negative.

Please solve it step by step so I can understand it too. xD