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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

Section 10-1

Formulas for cos (α ± β)

and sin (α ± β)

warm up
Warm – up:
  • What are the multiples of 30°, 45°, and 60°.
warm up1
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 up2
Warm-up:

What are the multiples of

warm up3
Warm-up:
  • Express each angle (a) as a sum and (b) as a difference of multiples of

4.

5.

formulas for cos
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 cos1
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
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 sin1
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
To summarize:
  • Sum and Difference Formulas for Cosine and Sine
the purpose
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
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
Example
  • Simplify the given expression:
    • cos 23° cos 67° + sin 23° sin 67°
    • sin 23° cos 67° + cos 23° sin 67°
example1
Example
  • Find the exact value of each expression.

sin 75° cos 165°

example2
Example
  • Simplify the given expression:
    • Sin (-t) cos 2t – cos (-t) sin 2t
example3
Example
  • Suppose that sin α = and sin β =

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