Textbook Question
Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
cot 30°
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Ch. 2 - Acute Angles and Right Triangles
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 3, Problem 6
Find one solution for each equation. Assume all angles involved are acute angles. cos(3θ + 11°) = sin( 7θ + 40°) 5 10
Verified step by step guidance1
Recall the co-function identity in trigonometry: \(\cos A = \sin B\) implies that either \(A = B\) or \(A = 90^\circ - B\) (considering acute angles).
Set up the first equation by equating the angles directly: \(3\theta + 11^\circ = 7\theta + 40^\circ\).
Solve the equation from step 2 for \(\theta\) by isolating \(\theta\) on one side.
Set up the second equation using the complementary angle relationship: \(3\theta + 11^\circ = 90^\circ - (7\theta + 40^\circ)\).
Solve the equation from step 4 for \(\theta\) by simplifying and isolating \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Sine and Cosine Functions
Sine and cosine functions are co-functions, meaning sin(α) = cos(90° - α). This identity allows us to rewrite equations involving sine and cosine in terms of each other, which is essential for solving equations like cos(3θ + 11°) = sin(7θ + 40°). Recognizing this relationship simplifies the problem by converting it into a single trigonometric function.
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5:33
Period of Sine and Cosine Functions
Solving Linear Trigonometric Equations
Solving equations like cos(A) = cos(B) or sin(A) = sin(B) involves finding angles A and B that satisfy the equality, considering the periodicity and symmetry of trigonometric functions. For acute angles, solutions are restricted to values between 0° and 90°, which narrows down possible solutions and helps identify valid angle measures.
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7:48Solving Linear Equations
Domain Restrictions for Acute Angles
Acute angles are angles between 0° and 90°. When solving trigonometric equations with this restriction, only solutions within this interval are valid. This constraint is crucial because trigonometric functions are periodic and can have multiple solutions, but the problem limits the solution set to acute angles, simplifying the solution process.
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Related Practice
Textbook Question
CONCEPT PREVIEW Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
csc 60°
Textbook Question
Concept Check Match each angle in Column I with its reference angle in Column II. Choices may be used once, more than once, or not at all. See Example 1. I. II. 5. A. 45° 6. 212° B. 60° 7. C. 82° 8. D. 30° 9. E. 38° 10. F. 32°
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Textbook Question
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1.
sin⁻¹ 0.30
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
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Textbook Question
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1.
sec 18°
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
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Textbook Question
CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.
Column I: 1.
cot 27°
Column II:
A. 88.09084757°
B. 63.25631605°
C. 1.909152433°
D. 17.45760312°
E. 0.2867453858
F. 1.962610506
G. 14.47751219°
H. 1.015426612
I. 1.051462224
J. 0.9925461516
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