Question

if sin = -1/2 ( 3π/2 < x < 2π ) then cosx/2 is ​

Answer 1

Answer:

Savannah, remember that there is a 5-12-13 right triangle here. That should come to your mind when you see 5 and 13.

We know that we can draw this triangle in the IV quadrant since 3π/2<x<2π. There is nothing magic in terms of π for 5-12-13 triangles; no π/4, π/6 or any nice round solutions.

We know that the cos(x) then = 12/13. (the edge of the same triangle adjacent to angle x in quadrant IV is positive and the hypotenuse of that triangle is +13.)

We know that cos(x+y) = cosxcosy - sinxsiny, so cos⁡(x+π/4) = cosxcos(π/4) - sinxsin(π/4) =

[sqrt(2)/2]*[cosx - sinx] = [sqrt(2)/2]*[12/13 - (- 5/13)] = 17*sqrt(2)/26.

Remember that cos(π/4) = sin(π/4)!