Lesson 52 Double Angle Half Angle Identities Math

Lesson 52 – Double Angle & Half Angle Identities Math 2 Honors - Santowski 12/17/2021 Math 2 Honors - Santowski 1

Fast Five n Graph the following functions and develop an alternative equation for the graphed function (i. e. Develop an identity for the given functions) n f(x) = 2 sin(x)cos(x) n g(x) = cos 2(x) – sin 2(x) 12/17/2021 IB Math SL - Santowski 2

Fast Five 12/17/2021 IB Math SL - Santowski 3

(A) Review n List the six new identities that we call the addition subtraction identities 12/17/2021 Math 2 Honors - Santowski 4

(A) Review n List the six new identities that we call the addition subtraction identities 12/17/2021 Math 2 Honors - Santowski 5

(B) Using the Addition/Subtraction Identities n We can use the addition/subtraction identities to develop new identities: n Develop a new identity for: n (a) sin(2 x) (b) cos(2 x) (c) tan(2 x) n n 12/17/2021 Math 2 Honors - Santowski 6

(B) Double Angle Formulas n sin(2 x) = 2 sin(x) cos(x) n cos(2 x) = cos 2(x) – sin 2(x) 12/17/2021 IB Math SL - Santowski 7

(B) Double Angle Formulas n Working with cos(2 x) = cos 2(x) – sin 2(x) n But recall the Pythagorean Identity where sin 2(x) + cos 2(x) = 1 So sin 2(x) = 1 – cos 2(x) And cos 2(x) = 1 – sin 2(x) n n So cos(2 x) = cos 2(x) – (1 – cos 2(x)) = 2 cos 2(x) - 1 And cos(2 x) = (1 – sin 2(x)) – sin 2(x) = 1 – 2 sin 2(x) 12/17/2021 IB Math SL - Santowski 8

(B) Double Angle Formulas n Working with cos(2 x) n cos(2 x) = cos 2(x) – sin 2(x) cos(2 x) = 2 cos 2(x) - 1 cos(2 x) = 1 – 2 sin 2(x) n n 12/17/2021 IB Math SL - Santowski 9

(C) Double Angle Identities Applications n (a) Determine the value of sin(2θ) and cos(2θ) if n (b) Determine the values of sin(2θ) and cos(2θ) if n (c) Determine the values of sin(2θ) and cos(2θ) if 12/17/2021 IB Math SL - Santowski 10

(C) Double Angle Identities Applications n Simplify the following expressions: 12/17/2021 IB Math SL - Santowski 11

(C) Double Angle Identities Applications n Write as a single function: 12/17/2021 IB Math SL - Santowski 12

(C) Double Angle Identities Applications n Use the double angle identities to prove that the following equations are identities: 12/17/2021 IB Math SL - Santowski 13

(D) Using the Double Angle Formulas n (a) If sin(x) = 21/29, where 0º < x < 90º, evaluate: (i) sin(2 x), (ii) cos(2 x), (iii) tan(2 x) n (b) SOLVE the equation cos(2 x) + cos(x) = 0 for 0º < x < 360º n (c) Solve the equation sin(2 x) + sin(x) = 0 for -180º < x < 540º. n (d) Write sin(3 x) in terms of sin(x) n (e) Write cos(3 x) in terms of cos(x) 12/17/2021 Math 2 Honors - Santowski 14

(E) Half Angle Identities n n n Start with the identity cos(2 x) = 1 – 2 sin 2(x) Isolate sin 2(x) this is called a “power reducing” identity Why? Now, make the substitution x = θ/2 and isolate sin(θ/2) Start with the identity cos(2 x) = 2 cos 2(x) - 1 Isolate cos 2(x) this is called a “power reducing” identity Why? Now, make the substitution x = θ/2 and isolate cos(θ/2) 12/17/2021 Math 2 Honors - Santowski 15

(E) Half Angle Identities n So the new half angle identities are: 12/17/2021 Math 2 Honors - Santowski 16

(F) Using the Half Angle Formulas n Prove the identity: 12/17/2021 Math 2 Honors - Santowski 17

(F) Using the Half Angle Formulas n (a) If sin(x) = 21/29, where 0º < x < 90º, evaluate: (i) sin(x/2), (ii) cos(x/2) n (b) develop a formula for tan(x/2) n (c) Use the half-angle formulas to evaluate: 12/17/2021 Math 2 Honors - Santowski 18

(F) Homework n HW n S 14. 5, p 920 -922, Q 8 -13, 20 -23, 25 -39 odds 12/17/2021 Math 2 Honors - Santowski 19