lesson 37 – addition & subtraction...
TRANSCRIPT
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Lesson 37 – Addition & Subtraction Identities
PreCalculus - Santowski
PreCalculus
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Fast Five
True or False (and justify your responses algebraically, graphically & numerically – but without the use of a calculator )
PreCalculus
€
(a) sin x +π4
⎛
⎝ ⎜
⎞
⎠ ⎟ = sin x( ) + sin
π4
⎛
⎝ ⎜ ⎞
⎠ ⎟
(b) cosπ2− x
⎛
⎝ ⎜
⎞
⎠ ⎟ = cos
π2
⎛
⎝ ⎜ ⎞
⎠ ⎟ +cos x( )
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(A) Six New Identities (GASP!!)
Here are six new identities that we call the addition & subtraction identities
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(B) Proving Cosine Subtraction Identity
We will prove
We will use our unit circle to do so ..... http://www.cut-the-knot.org/triangle/SinCosFormula.shtml
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(C) Proving Sine Addition Identity
We will prove
We will use right triangle trig to do so
http://www.cut-the-knot.org/triangle/SinCosFormula.shtml
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(D) Proving Tan Addition Identity
You will prove
You will use fundamental trig identities to do so
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(E) Using the Addition/Subtraction Identities
Evaluate each expression
1) sin (75°) using sin (45° + 30°) 2) sin (75°) using sin (120° – 45°) 3) cos (345°)
4)
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tan11π12
⎛
⎝ ⎜
⎞
⎠ ⎟
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(E) Using the Addition/Subtraction Identities
Determine the exact value of sin(15º)
Determine the exact value of cos(-195º)
Determine the exact value of
Determine the exact value of tan(255º)
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Pop Quiz
Find the exact value of
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sec13π12
⎛
⎝ ⎜
⎞
⎠ ⎟
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(E) Using the Addition/Subtraction Identities
Find each of the following numbers:
If
1) Evaluate sin (A + B) 2) Evaluate cos (A – B) 3) Evaluate tan (A + B)
4) If sin(a) = -4/5 for 180º<a<270º and if cos(b) = -5/13 for 90º<b<180º, evaluate tan(a+b)
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sinA =12
13, for
π2
< A <π and cosB = −8
17, for π <B <
3π2
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(E) Using the Addition/Subtraction Identities
Simplify the following:
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(1) cos 270°− x( )(2) sin x +
π2
⎛
⎝ ⎜
⎞
⎠ ⎟
(3) cos x +π( )
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(E) Using the Addition/Subtraction Identities
We can use the new identities to develop new identities:
Prove the following: (describe each identity from a transformations perspective as well as a unit circle perspective)
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(a) cos π + x( ) = −cosx
(b) cosπ2− x
⎛
⎝ ⎜
⎞
⎠ ⎟ = sinx
(c) sin π −θ ref( ) = sin θ ref( )
PreCalculus
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(E) Using the Addition/Subtraction Identities
Find the exact value of:
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(a) sin13°cos17°+ sin17°cos13°
(b) cos25°cos35°− sin25°sin35°
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(E) Using the Addition/Subtraction Identities
Prove the following:
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Solve for xER
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Solve for xER
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(E) Using the Addition/Subtraction Identities
We can use the new identities to develop new identities:
Develop a new identity for:
(a) sin(2x) (b) cos(2x) (c) tan(2x)
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Linear Combinations of Sine and Cosine
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(F) Homework
S14.4, p914, Q11,13,25,27,33-44all,67,77,78
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