Homework Log Fri 4/22 Lesson 8 – 4 Learning Objective: To use double & half angle identities to verify or find exact trig values Hw: #810 Pg. 509 #4, 8, 10, 12, 16, 24, 28, 32, 33 – 39 odd, 42, 48, 54

4/22/16 Lesson 8 – 4 Day 3 Double & Half Angle Identities Advanced Math/Trig

Learning Objective To use double or half angle identities to do the following: Find exact Trig values Rewrite Trig identities

1. Find the exact values of all the trig functions of 2𝜃 tan 𝜃 =−2 2 & sec 𝜃 >0 Not Special Angle but can use pythag identity 1+ 𝑡𝑎𝑛 2 𝜃= 𝑠𝑒𝑐 2 𝜃 So cos 𝜃 = 1 3 1+ 𝑡𝑎𝑛 2 𝜃 = sec 𝜃 Since tan 𝜃 = sin 𝜃 cos 𝜃 1+ (−2 2 ) 2 = sec 𝜃 tan 𝜃 cos 𝜃 = sin 𝜃 −2 2 1 3 = sin 𝜃 1+4(2) = sec 𝜃 −2 2 3 = sin 𝜃 9 = sec 𝜃 3 = sec 𝜃

#1 Cont’d Find the exact values of all the trig functions of 2𝜃 sin 𝜃 =− 2 2 3 cos 𝜃 = 1 3 tan 𝜃 =−2 2 =2 − 2 2 3 1 3 = −4 2 9 sin 2𝜃 =2 sin 𝜃 cos 𝜃 = 1 3 2 − − 2 2 3 2 cos 2𝜃 = 𝑐𝑜𝑠 2 𝜃− 𝑠𝑖𝑛 2 𝜃 = 1 9 − 8 9 =− 7 9 sec 2𝜃 = 1 cos 2𝜃 =− 9 7 =− 9 2 8 csc 2𝜃 = 1 sin 2𝜃 = 9 −4 2

#1 Cont’d Find the exact values of all the trig functions of 2𝜃 sin 𝜃 =− 2 2 3 cos 𝜃 = 1 3 tan 𝜃 =−2 2 = − 4 2 9 − 7 9 = −4 2 −7 = 4 2 7 tan 2𝜃 = sin 2𝜃 cos 2𝜃 = 7 2 8 cot 2𝜃 = 1 tan 2𝜃 = 7 4 2

2. Find the exact values of sin, cos, and tan of 3𝜋 8 3𝜋 8 in QI So cos 𝜃 >0 & sin 𝜃 >0 Use Half Angle Formulas! 3𝜋 8 = 𝜃 2 ? 3𝜋 8 =2𝜃 ? 3𝜋 16 =𝜃 6𝜋 8 =𝜃 cos 3𝜋 4 =− 2 2 NOT USEFUL 3𝜋 4 =𝜃 USEFUL

#2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8 cos 3𝜋 4 =− 2 2 cos 𝜃 >0 sin 𝜃 >0 sin 3𝜋 8 = sin 3𝜋 4 2 sin 𝜃 2 =± 1− cos 𝜃 2 = 1− − 2 2 2 = 2+ 2 2 2 = 2+ 2 4 = 2+ 2 2

#2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8 cos 3𝜋 4 =− 2 2 cos 𝜃 >0 sin 𝜃 >0 = 1+ − 2 2 2 cos 3𝜋 8 = cos 3𝜋 4 2 cos 𝜃 2 =± 1+ cos 𝜃 2 = 2− 2 2 2 = 2− 2 4 = 2− 2 2

#2 Cont’d Find the exact values of sin, cos, and tan of 3𝜋 8 = 2+ 2 2 2− 2 2 tan 3𝜋 8 = sin 3𝜋 8 cos 3𝜋 8 = 2+ 2 2− 2 ∙ 2− 2 2− 2 Multiply by denominator to get rid of big radical = 4−2 2− 2 = 2 2− 2 ∙ 2+ 2 2+ 2 = 2 (2+ 2 ) 4−2 Multiply by conjugate to get rid of radical = 2 2 +2 2 = 2 +1

Ticket Out the Door Determine the exact values of the sine, cosine, and tangent of 105 𝑜