5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for...

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Sum and Difference Identities for Cosine 5.3 Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities Mrs. Poland December 3, 2012 Saturday, December 1, 2012

Transcript of 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for...

Page 1: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Sum and Difference Identities for Cosine

5.3Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities

Mrs. PolandDecember 3, 2012

Saturday, December 1, 2012

Page 2: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Cosine of a Sum or Difference

Saturday, December 1, 2012

Page 3: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

FINDING EXACT COSINE FUNCTION VALUES Example 1(a and b)

Find the exact value of cos 15°.

Find the exact value of

Saturday, December 1, 2012

Page 4: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Example 1(c) FINDING EXACT COSINE FUNCTION VALUES

Find the exact value of cos 87°cos 93° – sin 87°sin 93°.

Saturday, December 1, 2012

Page 5: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Cofunction Identities

Similar identities can be obtained for a real number domain by replacing 90°

with

Saturday, December 1, 2012

Page 6: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Example 2 USING COFUNCTION IDENTITIES TO FIND θ

Find an angle that satisfies each of the following:

(a) cot θ = tan 25°

(b) sin θ = cos (–30°)

(c)

tan 25°tan 25°

Saturday, December 1, 2012

Page 7: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Example 3 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t

Suppose that and both s

and t are in quadrant II. Find cos(s + t).

Sketch an angle s in quadrant II such that Since let y = 3 and r = 5.

The Pythagorean theorem gives

Since s is in quadrant II, x = –4 and

Saturday, December 1, 2012

Page 8: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Example 3 FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

The Pythagorean theorem gives

Since t is in quadrant II, y = 5 and

Note

The values of cos s and sin t could also be found by using the

Pythagorean identities.

Sketch an angle t in quadrant II such that Since let x = –12 and r = 13.x

r,

Saturday, December 1, 2012

Page 9: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Classwork

14-5 ws (only cosine)

Saturday, December 1, 2012

Page 10: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Difference Identity for Cosine

Point Q is on the unit circle, so the coordinates of Q are (cos B, sin B).

The coordinates of S are (cos A, sin A).

The coordinates of R are (cos(A – B), sin (A – B)).

Saturday, December 1, 2012

Page 11: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Difference Identity for CosineSince the central angles SOQ and POR are equal, PR = SQ.

Using the distance formula, we have

PR=

SQ

=

Saturday, December 1, 2012

Page 12: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Difference Identity for CosineSquare both sides and clear parentheses:

PR=

SQ

Saturday, December 1, 2012

Page 13: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Rearrange the terms:

Saturday, December 1, 2012

Page 14: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Difference Identity for Cosine

Subtract 2, then divide by –2:

Saturday, December 1, 2012

Page 15: 5.3 Sum and Difference Identities for CosineSum Identity for Cosine To find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Sum Identity for CosineTo find a similar expression for cos(A + B) rewrite A + B as A – (–B) and use the identity for cos(A – B).

Cosine difference identity

Negative angle identities

Saturday, December 1, 2012


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