9/12/2015 hl math - santowski 1 lesson 24 – double angle & half angle identities hl math –...
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04/19/23 HL Math - Santowski 1
Lesson 24 – Double Angle & Half Angle Identities
HL Math – Santowski
04/19/23 HL Math - Santowski 2
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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