Lesson 24 Double Angle Half Angle Identities HL

Lesson 24 – Double Angle & Half Angle Identities HL Math – Santowski 1/9/2022 HL Math - Santowski 1

Fast Five n Graph the following functions on your TI-84 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) 1/9/2022 HL Math - Santowski 2

Fast Five 1/9/2022 HL Math - Santowski 3

(A) Review n List the six new identities that we call the addition subtraction identities 1/9/2022 HL Math - Santowski 4

(A) Review n List the six new identities that we call the addition subtraction identities 1/9/2022 HL Math - 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) if sin(2 x) = sin(x + x), then …. (b) if cos(2 x) = cos(x + x), then …. (c) if tan(2 x) = tan(x + x), then …… n n 1/9/2022 HL Math - 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) 1/9/2022 HL Math - 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) 1/9/2022 HL Math - Santowski 8

(B) Double Angle Formulas n Working with cos(2 x) n 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 sin(2 x) = 2 sin(x) cos(x) n 1/9/2022 HL Math - 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 1/9/2022 HL Math - Santowski 10

(C) Double Angle Identities Applications n Solve the following equations, making use of the double angle formulas for key substitutions: 1/9/2022 HL Math - Santowski 11

(C) Double Angle Identities Applications n Write as a single function: n State the period & amplitude of the single function 1/9/2022 HL Math - Santowski 12

(B) Double Angle Formulas n Working with cos(2 x) n 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 sin(2 x) = 2 sin(x) cos(x) n 1/9/2022 HL Math - Santowski 13

(C) Double Angle Identities Applications 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) 1/9/2022 HL Math - Santowski 14

(C) Double Angle Identities Applications n Simplify the following expressions: 1/9/2022 HL Math - Santowski 15

(C) Double Angle Identities Applications n Use the double angle identities to prove that the following equations are identities: 1/9/2022 HL Math - Santowski 16

(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) 1/9/2022 HL Math - Santowski 17

(E) Half Angle Identities n So the new half angle identities are: 1/9/2022 HL Math - Santowski 18

(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: 1/9/2022 HL Math - Santowski 19

(F) Using the Half Angle Formulas n Prove the identity: 1/9/2022 HL Math - Santowski 20

(F) Homework n HW n S 14. 5, p 920 -922, Q 8 -13, 20 -23, 25 -39 odds 1/9/2022 HL Math - Santowski 21