Use a half angle identity to find sin(x/2) if csc x=13 and pi/2<x<pi

Question

Use a half angle identity to find sin(x/2) if csc x=13 and pi/2<x<pi

Answer 1

Since our angle is in the second quadrant, we will be using the positive square root formula

sin(x/2) = √((1-cos(x))/2)

Now since csc(x) =13, this implies sin(x) = 1/13 since sin(x) = 1/csc(x)

So, if we draw the triangle in the second quadrant and label the sides given, then we can use the pythagorean theorem to find the missing side:

x² + 1² = 13²

x² + 1 = 169

x² = 168

x = - √(168) = -2√(42)

Which gives us a cosine value of:

cos(x) = -2√(42)/13 ≈ -0.997

Finally we can plug into our formula:

sin(x/2) = √((1-cos(x))/2)

= √((1-(-0.997)/2)

= √(1.997/2)

= √(0.9985)

≈ 0.9993