06/15/22 1 06/15/22 1 Lesson 37 – Addition & Subtraction Identities PreCalculus - Santowski PreCalculus
05/05/23 105/05/23 1
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
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(a) sinx+π4
⎛ ⎝ ⎜
⎞ ⎠ ⎟=sinx( )+sin
π4
⎛ ⎝ ⎜
⎞ ⎠ ⎟
(b) cosπ2−x
⎛ ⎝ ⎜
⎞ ⎠ ⎟=cosπ
2
⎛ ⎝ ⎜
⎞ ⎠ ⎟+cosx( )
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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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( )( )
( )( )
( )
( )BABABA
BABABA
BABABABABABA
ABBABAABBABA
tantan1tantantan
tantan1tantantan
sinsincoscoscossinsincoscoscos
cossincossinsincossincossinsin
+
=
+
=+
+==+
=+=+
PreCalculus
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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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( ) BABABA sinsincoscoscos =+
PreCalculus
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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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( ) ABBABA cossincossinsin +=+
PreCalculus
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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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( )BABABA
tantan1tantantan
+
=+
PreCalculus
(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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125sec π
PreCalculus
Pop Quiz
Find the exact value of
05/05/23 PreCalculus 9
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sec13π12
⎛ ⎝ ⎜
⎞ ⎠ ⎟
(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)
05/05/23 PreCalculus 10
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sinA=1213
, for π2< A<π and cosB=−
817
, for π <B<3π2
(E) Using the Addition/Subtraction Identities Simplify the following:
05/05/23 PreCalculus 11
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(1) cos270° −x( )
(2) sinx+π2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
(3) cosx+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) cosπ +x( )=−cosx
(b) cosπ2−x
⎛ ⎝ ⎜
⎞ ⎠ ⎟=sinx
(c) sinπ −θref( )=sinθref( )
PreCalculus
(E) Using the Addition/Subtraction Identities Find the exact value of:
05/05/23 PreCalculus 13
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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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( ) ( )( )
yxyxyx
xyyxyyx
cossinsintancot1 (b)
cossinsincoscos (a)+
=+
=+++
PreCalculus
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
05/05/23 19PreCalculus