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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)     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

⎛

⎝ ⎜

⎞

⎠ ⎟

(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

(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

(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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