sum and difference identities for cos, sin and tan function. Rotation matrix.

1. Sum and DifferenceSum and Difference
IdentitiesIdentities

2. Objective
To use the sum and
difference identities for
the sine, cosine, and
tangent functions
Page 1008

3. Sum and Difference Identities for the Cos FunctionSum and Difference Identities for the Cos Function
cos (cos (ΑΑ ++ ΒΒ) = cos) = cos ΑΑ coscos ΒΒ – sin– sin ΑΑ sinsin ΒΒ
cos (cos (ΑΑ –– ΒΒ) = cos) = cos ΑΑ coscos ΒΒ + sin+ sin ΑΑ sinsin ΒΒ
Sum and Difference Identities for the Sin FunctionSum and Difference Identities for the Sin Function
sin (sin (ΑΑ ++ ΒΒ) = sin) = sin ΑΑ coscos ΒΒ + cos+ cos ΑΑ sinsin ΒΒ
sin (sin (ΑΑ –– ΒΒ) = sin) = sin ΑΑ coscos ΒΒ – cos– cos ΑΑ sinsin ΒΒ
Sum and Difference Identities for the Tan FunctionSum and Difference Identities for the Tan Function
tan (tan (ΑΑ ++ ΒΒ) =) = tantan ΑΑ + tan+ tan ΒΒ
1 - tan1 - tan ΑΑ tantan ΒΒ
tan (tan (ΑΑ –– ΒΒ) =) =
tantan ΑΑ – tan– tan ΒΒ
1 + tan1 + tan ΑΑ tantan ΒΒ

4. Find cos 15°
cos 15° = cos (45° - 30°)
= cos 45° cos 30° + sin 45° sin 30°
= √2
2
∙
√3
2
+
√2
2
∙
1
2
=
√6
4
+
√2
4
=
√6 + √2
4

5. 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

6. Find tan 105°
tan 105° = tan ( 60° + 45°)
= tan 60° + tan 45°
1 – tan 60° tan 45°
= √3 + 1
1 - √3 ∙ 1
√3 + 1
1 - √3
= ∙ 1 + √3
1 + √3
= -2 - √3

7. Prove the identity :
cos ( x- ) = - cos x
cos x cos + sin x sin =
(-1 ) cos x + (0) sin x =
- cos x = - cos x

8. 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 = 25²
x = - √625 - 49
= -24
Thus, tan A = y/x = 7/24

9. In quadrant 1: 0º < B < 180º and
cos B = 8/17
R=17
Y
8² + y² = 17²
Y = √289 - 64
= 15
Thus, tan B = y/x = 15/8 X = 8
B

10. 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 Α + tan Β
1 - tan Α tan Β

11. 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 degrees counterclockwiseƟ
about the origin can be found by using
the rotation matrix:
Cos -SinɵCos -Sinɵ
ɵɵ
Sin Cosɵ ɵSin Cosɵ ɵ
X
Y
=
X’
Y’