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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

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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

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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)

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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:

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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

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