Half-Angle Identities 5.4C

Half-Angle Identities Sin 𝑥 2 = ± 1− cos 𝑥 2 Cos 𝑥 2 = ± 1+ cos 𝑥 2 Tan 𝑥 2 = ± 1− cos 𝑥 1+ cos 𝑥 Tan 𝑥 2 = 1− cos 𝑥 sin 𝑥 Tan 𝑥 2 = sin 𝑥 1+ cos 𝑥

Ex. Find the values between [0, 2𝜋] for which the following is true: Cos2x = sin2( 𝑥 2 ) Cos2x = (1− cos 𝑥) 2 2 Cos2x = 1− cos 𝑥 2 2cos2x = 1 – cos x 2( 1+ cos 2𝑥 2 )=1− cos 𝑥 1 + cos 2x = 1 – cos x

cos 2x + cos x = 0 cos2x – sin2x + cos x = 0 cos2x + cos2x – 1 + cos x = 0 2cos2x – 1 + cos x = 0 2cos2x + cos x – 1 = 0 (2cos x – 1)(cos x + 1) = 0 2cos x – 1 = 0 cos x + 1 = 0 2cos x = 1 cos x = -1 Cos x = 1 2 x = 180° x = 60°