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Section 10-1

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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

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

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

4.

5.

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)
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

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.
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

To summarize:
• Sum and Difference Formulas for Cosine and Sine
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.
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.
Example
• Simplify the given expression:
• cos 23° cos 67° + sin 23° sin 67°
• sin 23° cos 67° + cos 23° sin 67°
Example
• Find the exact value of each expression.

sin 75° cos 165°

Example
• Simplify the given expression:
• Sin (-t) cos 2t – cos (-t) sin 2t
Example
• Suppose that sin α = and sin β =

where π < β < < α < 2π. Find sin(α + β).