1. Sum and Difference Identities
14-4
Sum and Difference Identities
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up
Find each product, if possible.
1. AB
2. BA

3. Objectives
Evaluate trigonometric expressions by using sum and difference identities.
Use matrix multiplication with sum and difference identities to perform rotations.

4. Vocabulary
rotation matrix

5. Matrix multiplication and sum and difference identities are tools to find the coordinates of points rotated about the origin on a plane.

7. Example 1A: Evaluating Expressions with Sum and Difference Identities
Find the exact value of cos 15°.
Write 15° as the difference 45° – 30° because trigonometric values of 45° and 30° are known.
cos 15° = cos (45° – 30°)
Apply the identity for cos (A – B).
= cos 45° cos 30° + sin 45° sin 30°
Evaluate.
Simplify.

8. Example 1B: Proving Evaluating Expressions with Sum and Difference Identities
Find the exact value of
Write as the sum of
Apply the identity for tan (A + B).

9. Example 1B Continued
Evaluate.
Simplify.

10. Check It Out! Example 1a
Find the exact value of tan 105°.
Write 105° as the sum of 60° + 45° because trigonometric values of 60° and 45° are known.
tan 105°= tan(60° + 45°)
Apply the identity for tan (A + B).

11. Check It Out! Example 1a Continued
Evaluate. =
Simplify.

12. Check It Out! Example 1b
Find the exact value of each expression.
Write as the sum of
because
trigonometric values of
and
are known.
Apply the identity for sin (A – B).

13. Check It Out! Example 1b Continued
Find the exact value of each expression.
Evaluate.
Simplify.

14. Shifting the cosine function right  radians is equivalent to reflecting it across the x-axis. A proof of this is shown in Example 2 by using a difference identity.

15. Example 2: Proving Identities with Sum and Difference Identities
Prove the identity tan
tan
Choose the left-hand side to modify.
Apply the identity for tan (A + B).
Evaluate.
Simplify.

16. Check It Out! Example 2
Prove the identity
Apply the identity for cos A + B.
Evaluate.
= –sin x
Simplify.

17. Example 3: Using the Pythagorean Theorem with Sum and Difference Identities
Find cos (A – B) if sin A = with 0 < A < and if tan B = with 0 < B <
Step 1 Find cos A, cos B, and sin B.
Use reference angles and the ratio definitions sin A = and tan B = Draw a triangle in the appropriate quadrant and label x, y, and r for each angle.

18. In Quadrant l (Ql), 0° < A < 90° and sin A = .
Example 3 Continued
In Quadrant l (Ql), 0° < A < 90° and sin A = .
In Quadrant l (Ql),
0°< B < 90° and tan B = .
x = 4
y = 3 r B x
r = 3
y = 1 A

19. Example 3 Continued x2 + 12 = 32 32 + 42 = r2
y = 3
y = 1 A B x
x = 4
x = 32
= r2
Thus, cos A =
Thus, cos B = and sin B =
and sin A =

20. Example 3 Continued
Step 2 Use the angle-difference identity to find cos (A – B).
cos (A – B) = cosAcosB + sinA sinB
Apply the identity for cos (A – B).
Substitute for cos A, for cos B, and for sin B.
Simplify.
cos(A – B) =

21. Check It Out! Example 3
Find sin (A – B) if sinA = with 90° < A < 180° and if cosB = with 0° < B < 90°.
In Quadrant ll (Ql),
90< A < 180 and
sin A = .
In Quadrant l (Ql),
0< B < 90° and cos B =
x = 3 y
r = 5 B x
r = 5
y = 4 A

22. Check It Out! Example 3 Continued
r = 5
y = 4 A
x = 3 y
r = 5 B
x = 52
Thus, sin A =
and cos A =
52 – 32 = y2
Thus, cos B = and sin B =

23. Check It Out! Example 3 Continued
Step 2 Use the angle-difference identity to find sin (A – B).
sin (A – B) = sinAcosB – cosAsinB
Apply the identity for sin (A – B).
Substitute for sin A and sin B, for cos A, and for cos B.
sin(A – B) =
Simplify.

24. To rotate a point P(x,y) through an angle θ use a rotation matrix.
The sum identities for sine and cosine are used to derive the system of equations that yields the rotation matrix.

25. Example 4: Using a Rotation Matrix
Find the coordinates, to the nearest hundredth, of the points (1, 1) and (2, 0) after a 40° rotation about the origin.
Step 1 Write matrices for a 40° rotation and for the points in the question.
Rotation matrix.
Matrix of point coordinates.

26. Example 4 Continued
Step 2 Find the matrix product.
Step 3 The approximate coordinates of the points after a 40° rotation are (0.12, 1.41) and (1.53, 1.29).

27. Check It Out! Example 4
Find the coordinates, to the nearest hundredth, of the points in the original figure after a 60° rotation about the origin.
Step 1 Write matrices for a 60° rotation and for the points in the question.
R60° =
Rotation matrix.
S =
Matrix of point coordinates.

28. Check It Out! Example 4 Continued
Step 2 Find the matrix product.
R60° x s =

29. Lesson Quiz: Part I
1. Find the exact value of cos 75°
2. Prove the identity sin = cos θ
3. Find tan (A – B) for sin A = with 0 <A< and cos B = with 0 <B<

30. Lesson Quiz: Part II
4. Find the coordinates to the nearest hundredth of the point (3, 4) after a 60° rotation about the origin.