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Lesson 24 – Double Angle & Half Angle Identities

HL Math – Santowski

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Fast Five

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)

f(x) = 2sin(x)cos(x)

g(x) = cos2(x) – sin2(x)

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Fast Five

(A) Review

List the six new identities that we call the addition subtraction identities

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(A) Review

List the six new identities that we call the addition subtraction identities

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BA

BABA

BA

BABA

BABABA

BABABA

ABBABA

ABBABA

tantan1

tantantan

tantan1

tantantan

sinsincoscoscos

sinsincoscoscos

cossincossinsin

cossincossinsin

(B) Using the Addition/Subtraction Identities

We can use the addition/subtraction identities to develop new identities:

Develop a new identity for:

(a) if sin(2x) = sin(x + x), then …. (b) if cos(2x) = cos(x + x), then …. (c) if tan(2x) = tan(x + x), then ……

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(B) Double Angle Formulas

sin(2x) = 2 sin(x) cos(x)

cos(2x) = cos2(x) – sin2(x)

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(B) Double Angle Formulas

Working with cos(2x) = cos2(x) – sin2(x)

But recall the Pythagorean Identity where sin2(x) + cos2(x) = 1

So sin2(x) = 1 – cos2(x) And cos2(x) = 1 – sin2(x)

So cos(2x) = cos2(x) – (1 – cos2(x)) = 2cos2(x) - 1 And cos(2x) = (1 – sin2(x)) – sin2(x) = 1 – 2sin2(x)

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(B) Double Angle Formulas

Working with cos(2x)

cos(2x) = cos2(x) – sin2(x) cos(2x) = 2cos2(x) - 1 cos(2x) = 1 – 2sin2(x)

sin(2x) = 2 sin(x) cos(x)

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(C) Double Angle Identities - Applications (a) Determine the value of sin(2θ) and cos(2θ) if

(b) Determine the values of sin(2θ) and cos(2θ) if

(c) Determine the values of sin(2θ) and cos(2θ) if

2

and 4

1sin

2

3 and

6

5cos

2

3 and

5

2tan

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(C) Double Angle Identities - Applications Solve the following equations, making use of

the double angle formulas for key substitutions:

xxxxx

xxx

xxx

xxx

xxx

20for 0sincossin2cos (e)

20for 0cos2cos (d)

20for 0sin2sin (c)

2for cos2sin (b)

for cos2cos (a)

22

2

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(C) Double Angle Identities - Applications Write as a single function:

State the period & amplitude of the single function

(a) sinx cosx

(b) 2cos2 x 2sin2 x

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(B) Double Angle Formulas

Working with cos(2x)

cos(2x) = cos2(x) – sin2(x) cos(2x) = 2cos2(x) - 1 cos(2x) = 1 – 2sin2(x)

sin(2x) = 2 sin(x) cos(x)

(C) Double Angle Identities - Applications (a) If sin(x) = 21/29, where 0º < x < 90º, evaluate: (i) sin(2x),

(ii) cos(2x), (iii) tan(2x)

(b) SOLVE the equation cos(2x) + cos(x) = 0 for 0º < x < 360º

(c) Solve the equation sin(2x) + sin(x) = 0 for -180º < x < 540º.

(d) Write sin(3x) in terms of sin(x)

(e) Write cos(3x) in terms of cos(x)

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(C) Double Angle Identities - Applications Simplify the following expressions:

xxx

x

xx

x

xxx

xx

xx

x

sinsincos

2cos (f)

sincos

2cos (e)

12sin2cos (d) 2cos1

2sincos (c)

12cos (b) cos

2sin (a)

2

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(C) Double Angle Identities - Applications Use the double angle identities to prove that

the following equations are identities:

x

xx

x

xx

x

x

x

xx

x

cos

12coscos4

sin

2sin (c)

2sin1

2cos

2cos

2sin1 (b)

tan2cos1

2sin (a)

(E) Half Angle Identities

Start with the identity cos(2x) = 1 – 2sin2(x) Isolate sin2(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(2x) = 2cos2(x) - 1 Isolate cos2(x) this is called a “power reducing”

identity Why? Now, make the substitution x = θ/2 and isolate cos(θ/2)

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(E) Half Angle Identities

So the new half angle identities are:

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2

cos1

2cos

2

cos1

2sin

(F) Using the Half Angle Formulas (a) If sin(x) = 21/29, where 0º < x < 90º, evaluate: (i)

sin(x/2), (ii) cos(x/2)

(b) develop a formula for tan(x/2)

(c) Use the half-angle formulas to evaluate:

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225sin (e)

5.67cos (d) 5.112sin (c)

12

7cos (b)

12sin (a)

(F) Using the Half Angle Formulas Prove the identity:

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)sec(12

cos)sec(2 (b)

cos22

sin

2sin (a)

2

22

xx

x

x

xx

(F) Homework

HW

S14.5, p920-922, Q8-13,20-23, 25-39odds

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