MULTIPLE ANGLE & PRODUCT –TO-SUM IDENTITIES Section 5-5

I CAN  I can use double-angle and half-angle identities to evaluate trigonometric expressions and solve trigonometric equations.

DOUBLE ANGLE IDENTITIES  Sin 2θ = 2 sin θ cos θ  Cos 2θ = cos 2 θ – sin 2 θ  Cos 2θ = 2cos 2 θ – 1  Cos 2θ = 1 – 2sin 2 θ  Tan 2θ = 2tan θ / (1 - tan 2 θ)

EXAMPLE  Evaluate Expressions involving Double Angles  If Sin θ = -7/25 on the interval (π, 3π/2), find sin 2θ, cos 2θ, tan 2θ.  Sin = y/r - we need to find x  x 2 = 25 2  x = -24

EXAMPLE  Evaluate Expressions involving Double Angles  If Sin θ = -7/25 on the interval (π, 3π/2), find sin 2θ, cos 2θ, tan 2θ.  Sin 2θ = 2sin θcos θ  2(-7/25)(-24/25)  336/625

EXAMPLE  Evaluate Expressions involving Double Angles  If Sin θ = -7/25 on the interval (π, 3π/2), find sin 2θ, cos 2θ, tan 2θ.  cos 2θ = 2cos 2 θ – 1  2(-24/25)  527/625

EXAMPLE  Evaluate Expressions involving Double Angles  If Sin θ = -7/25 on the interval (π, 3π/2), find sin 2θ, cos 2θ, tan 2θ.  tan 2θ = 2tan θ/(1 – tan 2 θ)  2(7/24)/(1 - (7/24) 2  336/527

PRACTICE  Evaluate Expressions involving Double Angles  If cos θ = 3/5 on the interval (0, π/2), find sin 2θ, cos 2θ, tan 2θ.  Cos = x/r - we need to find y  y 2 = 5 2  x = 4

PRACTICE  Evaluate Expressions involving Double Angles  If cos θ = 3/5 on the interval (0, π/2), find sin 2θ, cos 2θ, tan 2θ.  Sin 2θ = 2sin θcos θ  2(4/5)(3/5)  24/25

PRACTICE  Evaluate Expressions involving Double Angles  If cos θ = 3/5 on the interval (0, π/2), find sin 2θ, cos 2θ, tan 2θ.  cos 2θ = 2cos 2 θ – 1  2(3/5)  -7/25

PRACTICE  Evaluate Expressions involving Double Angles  If cos θ = 3/5 on the interval (0, π/2), find sin 2θ, cos 2θ, tan 2θ.  tan 2θ = 2tan θ/(1 – tan 2 θ)  2(4/3)/(1 - (4/3) 2 )  -24/7

HALF ANGLE IDENTITIES  Sin θ/2 = +√(1 - cos θ)/ 2  Cos θ/2 = +√(1 + cos θ)/ 2  Tan θ/2 = +√(1 - cos θ)/ (1 + cosθ)  Tan θ/2 =(1 - cos θ)/ sin θ  Tan θ/2 = sin θ / (1 + cos θ)

EXAMPLE  Evaluate an expressions involving half angle  Find the exact value of Cos 112.5˚.  Double the angle and write it like:  Cos 225˚/2  + √(1+ cos 225)/2  + √(1+ √2/2)/2  + √(2 - √2)/4

PRACTICE  Evaluate an expressions involving half angle  Find the exact value of Sin 75˚.  Sin 150˚/2  + √(1- cos 150)/2  + √(1- √3/2)/2  + √(2 - √3)/2

PRACTICE  Evaluate an expressions involving half angle  Find the exact value of Tan 7π/12.  Tan 7π/6  (1- cos 7π/6)/Sin 7π/6  (1- -√3/2)/(-1/2)  -2 - √3

ASSIGNMENT  HW pg 352: 1- 7 odd, all