We can obtain formulas involving half-angles from the double-angle formulas. To find an

Chapter 9, Problem 9.2.24

(choose chapter or problem)

We can obtain formulas involving half-angles from the double-angle formulas. To find an expression for sin 1 2 v, we solve cos 2u = 1 2 sin2 u for sin2 u to obtain sin2 u = 1 cos 2u 2 . Taking the square root gives: sin u = 1 cos 2u 2 , and finally we make the substitution u = 1 2 v to get: sin 1 2 v = 1 cos v 2 . (a) Show that cos 1 2 v = 1 + cos v 2 .(b) Show that tan 1 2 v = 1 cos v 1 + cos v . The sign in these formulas depends on the value of v/2. If v/2 is in quadrant I then the sign of sin v/2 is +, the sign of cos v/2 is +, and the sign of tan v/2 is +. (c) If v/2 is in quadrant II find the signs for sin v/2, for cos v/2 and for tan v/2. (d) If v/2 is in quadrant III find the signs for sin v/2, for cos v/2 and for tan v/2. (e) If v/2 is in quadrant IV find the signs for sin v/2, for cos v/2 and for tan v/2.