Download - 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

Cosine of a Sum or Difference


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

Find the exact value of cos 15°.

Find the exact value of


Example 1(c) FINDING EXACT COSINE FUNCTION VALUES

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


Cofunction Identities

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

with


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°


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


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,


Classwork

14-5 ws (only cosine)


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)).


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

Using the distance formula, we have

PR=

SQ

=


Difference Identity for CosineSquare both sides and clear parentheses:

PR=

SQ


Rearrange the terms:


Difference Identity for Cosine

Subtract 2, then divide by –2:


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
