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Does the sin(75) =sin(45)+sin(30) ?

Check it out. Does the sin(75) =sin(45)+sin(30) ?. 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° ).

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Does the sin(75) =sin(45)+sin(30) ?

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  1. Check it out Does the sin(75) =sin(45)+sin(30) ?

  2. 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.

  3. 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).

  4. Example 1B Continued Evaluate. Simplify.

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

  6. 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).

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

  8. 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.

  9. 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

  10. 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

  11. 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) =

  12. 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°.

  13. 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

  14. 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.

  15. 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

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