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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. - PowerPoint PPT Presentation
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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°.
30° 45° 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 3
,4
,6
Warm-up:
Express each angle (a) as a sum and (b) as a difference of multiples of
4.
5.
3,
4,
6
12
19
12
Formulas for cos (α ± β)
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.
AOB
Formulas for cos (α ± 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
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.
2
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
sincoscossinsin
sinsincoscoscos
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.
2sin
2sin2coscos
2cos
2cos2coscos
2sin
2cos2sinsin
2cos
2sin2sinsin
yxyxyx
yxyxyx
yxyxyx
yxyxyx
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).
