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14-4. Sum and Difference Identities. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Find each product, if possible. 1. AB. 2. BA. Objectives. Evaluate trigonometric expressions by using sum and difference identities.

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14-4

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  1. 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.

  6. 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.

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

  8. Example 1B Continued Evaluate. Simplify.

  9. 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).

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

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

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

  13. 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.

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

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

  16. Find cos (A – B) if sin A = with 0 < A < and if tan B = with 0 < 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. Example 3: Using the Pythagorean Theorem with Sum and Difference Identities Step 1 Find cos A, cos B, and sin B.

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

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

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

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

  21. x2 + 42 = 52 r = 5 52– 32 = y2 y r = 5 y = 4 A B Thus, cos B = and sin B = x Thus, sin A = x = 3 and cos A = Check It Out! Example 3 Continued

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

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

  24. 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.

  25. 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).

  26. R60° = S = 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. Rotation matrix. Matrix of point coordinates.

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

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

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

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