Find the exact value of cos β/2 using the half-angle formulas.

Question

tan β = 28/45, 0 <  β < pi/2

Answer 1

Find the exact value of cos β/2 using the half-angle formulas.

tan β = 28/45, 0 < β < pi/2

First, draw a right triangle in the first quadrant with β near the origin, and a vertical leg of 28, horizontal leg of 45. This will allow you to find that the hypotenuse is 45. You know that the angle is in the first quadrant because of the restriction on β. Now you can tell that cos β = 45/53. This also tells you'll be using the positive version of the cosine half angle formula, because cosine is positive in the first quadrant.

cos (β/2) = √((1 + cos β)/2)

So cos (β/2) = √((1 + 45/53)/2)

= √((98/53)/2) = √(98/106) = √(49/53) = 7/ √(53) = (7√(53))/(53)

I would say "7 root 53 over 53."

Answer 2

Draw and label a diagram.

Determine sine and cosine

Use appropriate half-angle formula