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Section 10-1

Section 10-1. Formulas for cos (α ± β) and sin (α ± β). Warm – up:. What are the multiples of 30°, 45°, and 60°. Warm – up:. Express each angle (a) as a sum and (b) as a difference of multiples of 30°, 45°, or 60°. 1. 255° 2. 195° 3. 345°. Warm-up:. What are the multiples of.

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Section 10-1

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  1. Section 10-1 Formulas for cos (α ± β) and sin (α ± β)

  2. Warm – up: • What are the multiples of 30°, 45°, and 60°.

  3. Warm – up: Express each angle (a) as a sum and (b) as a difference of multiples of 30°, 45°, or 60°. 1. 255° 2. 195° 3. 345°

  4. Warm-up: What are the multiples of

  5. Warm-up: • Express each angle (a) as a sum and (b) as a difference of multiples of 4. 5.

  6. To find a formula for cos (α - β), let A and B be points on the unit circle with coordinates as shown in the diagram at the right. Then the measure of is α – β. The distance AB can be found by using either the law of cosines or the distance formula. Examine both methods on p. 369. Formulas for cos (α ± β)

  7. Formulas for cos (α ± β) cos (α - β) = cos α cos β + sin α sin β • Therefore, • To obtain a formula for cos (α + β), we can use the formula for cos (α - β) and replace β with – β. Recall that cos (- β) = cos β and sin (- β) = - sin β. • cos (α + β) = cos α cos β - sin α sin β • Therefore, cos (α + β) = cos α cos β - sin α sin β

  8. Formulas for sin (α ± β) • To find formulas for sin (α + β), we use the cofunction relationship sin Θ = cos (recall… sin Θ = cos (90° - Θ)) • Look at derivation of formula on p. 370.

  9. Formulas for sin (α ± β) • Therefore, • And, sin (α + β) = sin α cos β + cos α sin β sin (α - β) = sin α cos β - cos α sin β

  10. To summarize: • Sum and Difference Formulas for Cosine and Sine

  11. The purpose… • There are two main purposes for the addition formulas: finding exact values of trigonometric expressions and simplifying expressions to obtain other identities. • The sum and difference formulas can be used to verify many identities that we have seen, such as sin (90° - Θ) = cos Θ, and also to derive new identities.

  12. Rewriting a Sum or Difference as a Product • Sometimes a problem involving a sum can be more easily solved if the sum can be expressed as a product.

  13. Example • Simplify the given expression: • cos 23° cos 67° + sin 23° sin 67° • sin 23° cos 67° + cos 23° sin 67°

  14. Example • Find the exact value of each expression. sin 75° cos 165°

  15. Example • Simplify the given expression: • Sin (-t) cos 2t – cos (-t) sin 2t

  16. Example • Suppose that sin α = and sin β = where π < β < < α < 2π. Find sin(α + β).

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