Sum and Sum and Difference Difference IdentitiesIdentities

ObjectiveTo use the sum and

difference identities for the sine, cosine, and tangent

functionsPage 1008

Sum and Difference Identities for the Cos Function Sum and Difference Identities for the Cos Function cos (cos ( + + ) = cos ) = cos cos cos – sin – sin sin sin cos (cos ( – – ) = cos ) = cos cos cos + sin + sin sin sin

Sum and Difference Identities for the Sin FunctionSum and Difference Identities for the Sin Functionsin (sin (+ + ) = sin ) = sin cos cos + cos + cos sin sin sin (sin ( – – ) = sin ) = sin cos cos – cos – cos sin sin

Sum and Difference Identities for the Tan FunctionSum and Difference Identities for the Tan Function

tan (tan ( + + ) =) = tan tan + tan + tan 1 - tan 1 - tan tan tan

tan (tan ( – – ) =) = tan tan – tan – tan 1 + tan 1 + tan tan tan

Find cos 15°cos 15° = cos (45° - 30°)

= cos 45° cos 30° + sin 45° sin 30°

= √2 2

∙ √3 2

+ √2 2

∙ 12

=√6 4 + √2

4

= √6 + √2 4

Find sin 15°sin 15° = sin (45° - 30°)

= sin 45° sin 30° - cos 45° sin 30°= √2

2∙ √3

2- √2

2∙ 1

2= √6

4- √2

4

= √6 - √2 4

Find tan 105°tan 105° = tan ( 60° + 45°)

= tan 60° + tan 45°1 – tan 60° tan 45°

= √3 + 11 - √3 1∙ √3 + 11 - √3

= ∙ 1 + √31 + √3

= -2 - √3

Prove the identity :

cos ( x- ) = - cos x

cos x cos + sin x sin =

(-1 ) cos x + (0) sin x =

- cos x = - cos x

Find tan ( A+B ) if sin A = -7/25 with 180º < A < 270º and if cos B = 8/17 with 0º < B < 180º

Step 1 : Find tan A and tan B Use reference angles and the ratio definitions sin A = y/r and cos B = x/r .

In quadrant 3 : 180º < A < 270º and sin A = -7/25

x

Y = -7

R = 25

A

x² + (-7)² = 25²x = - √625 - 49 = -24Thus, tan A = y/x = 7/24

In quadrant 1: 0º < B < 180º and cos B = 8/17

R=17

Y8² + y² = 17²Y = √289 - 64 = 15Thus, tan B = y/x = 15/8 X = 8

B

Step 2 : Use the angle-sum identify to find tan( A + B ).

Tan( A + B ) = 7 + 15 = 24 8 1- (7/24)(15/8)

= 416/87

tan + tan 1 - tan tan

Using a Rotation Matrix If P(x , y) is any point in a plane, then the coordinates P’(x’ , y’) of the image after a rotation of Ɵ degrees counterclockwise about the origin can be found by using the rotation matrix:

Cos ɵ -Sin ɵCos ɵ -Sin ɵ

Sin ɵ Cos ɵSin ɵ Cos ɵ

X

Y=

X’

Y’