Download Activity 6.6.3 Angle Sum Formulas

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Activity 6.6.3 Angle Sum and Difference Formulas (+)
sin(𝑎 + 𝑏) = 𝑠𝑖𝑛𝑎 ∙ 𝑐𝑜𝑠𝑏 + 𝑐𝑜𝑠𝑎 ∙ 𝑠𝑖𝑛𝑏
cos(𝑎 + 𝑏) = cos(𝑎) cos(𝑏) − sin(𝑎) sin⁡(𝑏)
tan(𝑎 + 𝑏) = ⁡
𝑡𝑎𝑛𝑎 + 𝑡𝑎𝑛𝑏
1 − 𝑡𝑎𝑛𝑎 ∙ 𝑡𝑎𝑛𝑏
sin(𝑎 − 𝑏) = 𝑠𝑖𝑛𝑎 ∙ 𝑐𝑜𝑠𝑏 − 𝑐𝑜𝑠𝑎 ∙ 𝑠𝑖𝑛𝑏
cos(𝑎 − 𝑏) = cos(𝑎) cos(𝑏) + sin(𝑎) sin⁡(𝑏)
tan(𝑎 − 𝑏) = ⁡
𝑡𝑎𝑛𝑎 − 𝑡𝑎𝑛𝑏
1 + 𝑡𝑎𝑛𝑎 ∙ 𝑡𝑎𝑛𝑏
Do not use a calculator for this activity.
1. Use 120° and 45° to find exact values for the following trigonometric functions.
a. sin(165°)
b. cos(165°)
c. tan(165°)
Activity 6.6.3
Connecticut Core Algebra 2 Curriculum Version 3.0
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2. Find exact values for the following:
a. sin(75°)
b. cos(75°)
c. tan(75°)
3. Show that you used an angle sum or difference formula to find exact values for the following:
7𝜋
a. 𝑠𝑖𝑛 12
Activity 6.6.3
−𝜋
b. 𝑐𝑜𝑠 12
Connecticut Core Algebra 2 Curriculum Version 3.0
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c. 𝑡𝑎𝑛105°
4. Derive the double angle formula for sine and cosine. Note sin(2a) that can be written sin(a + a)
a. Use the angle sum formula to expand sin(2a) using the angle sum formula and simplify.
b. Use the angle sum formula to expand cos(2a) and simplify.
Activity 6.6.3
Connecticut Core Algebra 2 Curriculum Version 3.0
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c. Write the double angle formula for sine and the double angle formula for cosine:
d. Verify an instance of the double angle formula for cosine by evaluating each side of the
equation and showing it is true:
cos(60°) = cos2(30°) – sin2(30°)
5. Simplify the following, give exact values. Do this without a calculator.
a. sin(10°⁡) ∙ cos(40°⁡) − cos⁡(10°⁡) ∙ sin⁡(40°⁡)
9𝜋
11𝜋
9𝜋
11𝜋
b. cos( 8 ) cos( 24 ) + sin( 8 ) sin( 24 )
Activity 6.6.3
Connecticut Core Algebra 2 Curriculum Version 3.0
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c. 2 sin°15·cos15°
5𝜋
5𝜋
5𝜋
5𝜋
d. cos2( 8 ) − sin2( 8 )
e. cos2( 8 ) + sin2( 8 )
𝜋
5𝜋
𝜋
5𝜋
f. sin (12⁡) ∙ cos (⁡ 12 ) + cos⁡(12⁡) ∙ sin⁡( 12 ⁡)
𝑡𝑎𝑛25°+𝑡𝑎𝑛200°
g. 1−𝑡𝑎𝑛25°∙𝑡𝑎𝑛200°
Activity 6.6.3
Connecticut Core Algebra 2 Curriculum Version 3.0