6. Trigonometric Identities and More Equations
Sum and Difference Identities
6. Trigonometric Identities and More Equations
Sum and Difference Identities - Video Tutorials & Practice Problems
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concept
Sum and Difference of Sine & Cosine
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6mPlay a video:
2
example
Example 1
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3mPlay a video:
3
Problem
ProblemFind the exact value of the expression.
cos105°
A
42−6
B
46−2
C
22−6
D
44
4
Problem
ProblemFind the exact value of the expression.
sin15°
A
42−6
B
46−2
C
22−6
D
44
5
Problem
ProblemFind the exact value of the expression.
cos125π
A
42−6
B
46−2
C
22−6
D
44
6
example
Example 2
Video duration:
1mPlay a video:
7
Problem
ProblemFind the exact value of the expression.
cos80°cos20°+sin80°sin20°
A
−21
B
0
C
21
D
23
8
example
Example 3
Video duration:
2mPlay a video:
9
Problem
ProblemExpand the expression using the sum & difference identities and simplify.
sin(−θ−2π)
A
−sinθ−cosθ
B
0
C
−sinθ
D
−cosθ
10
concept
Sum and Difference of Tangent
Video duration:
4mPlay a video:
11
Problem
ProblemFind the exact value of the expression.
tan105°
A
0
B
3−2
C
−2−3
D
23+2
12
Problem
ProblemExpand the expression using the sum & difference identities and simplify.
tan(−θ−2π)
A
−tanθ
B
tanθ
C
−cotθ
D
cotθ
13
concept
Verifying Identities with Sum and Difference Formulas
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2mPlay a video:
14
example
Example 4
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3mPlay a video:
15
example
Example 5
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3mPlay a video:
16
example
Example 6
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6mPlay a video:
17
concept
Evaluating Sums and Differences Given Conditions
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6mPlay a video:
18
example
Example 7
Video duration:
5mPlay a video:
19
Problem
ProblemFind cos(a+b) given cosa=21, sinb=21, & a is in Q IV and b is in Q II.
A
0
B
43
C
1
D
−23
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PRACTICE PROBLEMS AND ACTIVITIES (102)
- Be sure that you've familiarized yourself with the first set of formulas presented in this section by working ...
- Use 105° = 135° - 30° to find the exact value of 105°.
- Find the exact value of each expression. (Do not use a calculator.)cos(-15°)
- Find the exact value of each expression.sin 255°
- Find the exact value of each expression. (Do not use a calculator.)cos 105° (Hint: 105° = 60° + 45°)
- Find the exact value of each expression.tan 285°
- Find the exact value of each expression. (Do not use a calculator.)cos π/12
- Find the exact value of each expression.sin (13π/12)
- Find the exact value of each expression. (Do not use a calculator.)cos (-7π/12)
- Find the exact value of each expression.tan (5π/12)
- Find the exact value of each expression. (Do not use a calculator.)cos (7π/9) cos (2π/9) - sin (7π/9) si...
- Find the exact value of each expression.sin (π/12)
- Write each function value in terms of the cofunction of a complementary angle.sin 15°
- Find the exact value of each expression.sin (- 5π/12)
- Write each function value in terms of the cofunction of a complementary angle.sin (2π/5)
- Find the exact value of each expression.tan (-7π/12)
- Write each function value in terms of the cofunction of a complementary angle.sin 142° 14'
- Find the exact value of each expression.sin (-13π/12)
- Write each function value in terms of the cofunction of a complementary angle.cot (9π/10)
- Find the exact value of each expression. See Example 1.sin 40° cos 50° + cos 40° sin 50°
- Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.sin x = 3/5, c...
- Write each function value in terms of the cofunction of a complementary angle.tan 174° 03'
- Find the exact value of each expression. See Example 1.sin 5π/9 cos π/18 - cos 5π/9 sin π/18 .
- Use the given information to find sin(x + y).sin y = - 2/3 , cos x = - 1/5, x in quadrant II, y in quadrant II...
- Write each function value in terms of the cofunction of a complementary angle.sin 98.0142°
- Find the exact value of each expression. See Example 1.[tan 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]
- Use the given information to find sin(x + y).cos x = 2/9, sin y = - 1, x in quadrant IV, y in quadrant III
- Use identities to fill in each blank with the appropriate trigonometric function name.sin 2π/3 = _____ (- π/6...
- Find the exact value of each expression. See Example 1.[tan 5π/12 + tan π/4]/[1 - tan 5π/12 tan π/4]
- Use identities to fill in each blank with the appropriate trigonometric function name.____ 72° = cot 18°
- Write each function as an expression involving functions of θ or x alone. See Example 2.cos(θ - 30°)
- Use identities to fill in each blank with the appropriate trigonometric function name.tan 24° = 1/ _____ 66°
- Write each function as an expression involving functions of θ or x alone. See Example 2.cos(45° - θ)
- Find one value of θ or x that satisfies each of the following.tan θ = cot(45° + 2θ)
- Find one value of θ or x that satisfies each of the following.sin θ = cos(2θ + 30°)
- Write each function as an expression involving functions of θ or x alone. See Example 2.sin(45° + θ)
- Find one value of θ or x that satisfies each of the following.sec x = csc (2π/3)
- Find one value of θ or x that satisfies each of the following.cos x = sin (π/12)
- Write each function as an expression involving functions of θ or x alone. See Example 2.tan (π/4 + x)
- Find one value of θ or x that satisfies each of the following.cot(θ - 10°) = tan(2θ - 20°)
- Write each function as an expression involving functions of θ or x alone. See Example 2.sin (3π/4 - x)
- Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function ...
- Write each function as an expression involving functions of θ or x alone. See Example 2.tan(180° + θ)
- Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function ...
- Write each function as an expression involving functions of θ or x alone. See Example 2.sin(π + x)
- Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function ...
- Express each function as a trigonometric function of x. See Example 5.cos 3x
- Use the identities for the cosine of a sum or difference to write each expression as a trigonometric function ...
- Express each function as a trigonometric function of x. See Example 5.cos 4x
- Find cos(s + t) and cos(s - t).cos s = - 8/17 and cos t = - 3/5, s and t in quadrant III
- Find cos(s + t) and cos(s - t).sin s = 2/3 and sin t = -1/3, s in quadrant II and t in quadrant IV
- Find cos(s + t) and cos(s - t).cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV
- Match each expression in Column I with its equivalent expression in Column II.sin 60° cos 45° - cos 60° sin 45...
- Verify that each equation is an identity (Hint: cos 2x = cos(x + x).) cos( π/2 + x) = -sin x
- Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)cos 2x = cos² x - sin² x
- Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)cos 2x = 1 - 2 sin² x
- Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)cos 2x = (cot² x - 1)/(cot² x + 1)
- Match each expression in Column I with its equivalent expression in Column II.(tan (π/3) - tan (π/4))/(1 + tan...
- Be sure that you've familiarized yourself with the first set of formulas presented in this section by working ...
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find t...
- In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. sin 195°
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find t...
- In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. 5𝝅 tan -----...
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find t...
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find t...
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write ...
- Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write ...
- In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α 3 𝝅 12 𝝅 ...
- In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) 3 𝝅 12 ...
- In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) 3 𝝅 12 ...
- In Exercises 35–38, find the exact value of the following under the given conditions: a. sin(α + β) ...
- In Exercises 35–38, find the exact value of the following under the given conditions: c. tan(α + β) ...
- In Exercises 35–38, find the exact value of the following under the given conditions: d. sin 2α ...
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 3 5 sin...
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 3 5 sin...
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 8 1 cos...
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 8 1 cos...
- In Exercises 57–64, find the exact value of the following under the given conditions: c. tan (α + β) 5 𝝅 3 ...
- In Exercises 57–64, find the exact value of the following under the given conditions: b. sin (α + β) 5 𝝅 3 ...
- In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β...
- Use the given information to find cos(x - y).sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III
- Use the given information to find tan(x + y).sin y = - 2/3, cos x = -1/5, x in quadrant II, y in quadrant III
- Use the given information to find the quadrant of x + y.sin y = - 2/3, cos x = -1/5 , x in quadrant II, y in q...
- Use the given information to cos(x - y).cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
- Use the given information to find tan(x + y).cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quadrant III
- Use the given information to find the quadrant of x + y.cos x = 2/9, sin y = -1/2, x in quadrant IV, y in quad...
- Verify that each equation is an identity.sin(x + y) + sin(x - y) = 2 sin x cos y
- Verify that each equation is an identity. See Example 4.tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x ...
- Verify that each equation is an identity.sin(s + t)/cos s cot t = tan s + tan t
- Verify that each equation is an identity.sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)
- Verify that each equation is an identity.(tan(α + β) - tan β)/(1 + tan(α + β) tan β) = tan α
- Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of ...
- Use the result from Exercise 80 to find the acute angle between each pair of lines. (Note that the tangent of ...
- Use the given information to find sin(s + t). See Example 3.sin s = 3/5 and sin t = -12/13, s in quadrant I a...
- Use the given information to find tan(s + t). See Example 3.sin s = 3/5 and sin t = -12/13, s in quadrant I a...
- Use the given information to find the quadrant of s + t. See Example 3.sin s = 3/5 and sin t = -12/13, s in q...
- Use the given information to find sin(s + t). See Example 3.cos s = -15/17 and sin t = 4/5, s in quadrant II a...
- Use the given information to find tan(s + t). See Example 3.cos s = -15/17 and sin t = 4/5, s in quadrant II ...
- Use the given information to find the quadrant of s + t. See Example 3.cos s = - 15/17 and sin t = 4/5, s in q...
- Use the given information to find sin(s + t). See Example 3.cos s = -1/5 and sin t = 3/5, s and t in quadrant...
- Use the given information to find tan(s + t). See Example 3.cos s = -1/5 and sin t = 3/5, s and t in quadrant...
- Use the given information to find the quadrant of s + t. See Example 3.cos s = -1/5 and sin t = 3/5, s and t ...