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4/27/14 1 4/27/14 1 Lesson 37 – Addition & Subtraction Identities PreCalculus - Santowski PreCalculus

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### Transcript of Lesson 37 – Addition & Subtraction... 4/27/14 1 4/27/14 1

Lesson 37 – Addition & Subtraction Identities

PreCalculus - Santowski

PreCalculus 4/27/14 2

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( ) 4/27/14 3

(A) Six New Identities (GASP!!)

Here are six new identities that we call the addition & subtraction identities

4/27/14 3 PreCalculus 4/27/14 4

(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

4/27/14 4 PreCalculus 4/27/14 5

We will prove

We will use right triangle trig to do so

http://www.cut-the-knot.org/triangle/SinCosFormula.shtml

4/27/14 5 PreCalculus 4/27/14 6

You will prove

You will use fundamental trig identities to do so

4/27/14 6 PreCalculus Evaluate each expression

1) sin (75°) using sin (45° + 30°)   2) sin (75°) using sin (120° – 45°)   3) cos (345°)

4)

4/27/14 PreCalculus 7

tan11π12

⎝ ⎜

⎠ ⎟ 4/27/14 8

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)

4/27/14 8 PreCalculus Pop Quiz

Find the exact value of

4/29/14 PreCalculus 9

sec13π12

⎝ ⎜

⎠ ⎟ 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)

4/27/14 PreCalculus 10

sinA =12

13,  for

π2

< A <π    and     cosB = −8

17,  for  π <B <

3π2 Simplify the following:

4/27/14 PreCalculus 11

(1)     cos 270°− x( )(2)     sin x +

π2

⎝ ⎜

⎠ ⎟

(3)     cos x +π( ) 4/27/14 12

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)

4/27/14 12

(a)     cos π + x( ) = −cosx

(b)     cosπ2− x

⎝ ⎜

⎠ ⎟ = sinx

(c)     sin π −θ ref( ) = sin θ ref( )

PreCalculus Find the exact value of:

4/27/14 PreCalculus 13

(a)       sin13°cos17°+ sin17°cos13°

(b)     cos25°cos35°− sin25°sin35° 4/27/14 14

Prove the following:

4/27/14 14 PreCalculus Solve for xER

4/29/14 PreCalculus 15 Solve for xER

4/29/14 PreCalculus 16 4/27/14 17

We can use the new identities to develop new identities:

Develop a new identity for:

(a) sin(2x)   (b) cos(2x)   (c) tan(2x)

4/27/14 17 PreCalculus Linear Combinations of Sine and Cosine

4/27/14 PreCalculus 18 4/27/14 19

(F) Homework

S14.4, p914, Q11,13,25,27,33-44all,67,77,78

4/27/14 19 PreCalculus