Textbook Question
Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. 112° 15'
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Ch. 1 - Trigonometric Functions
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 2, Problem 54
Determine whether each statement is possible or impossible. See Example 4. sin θ = 3
Verified step by step guidance1
Recall the range of the sine function: for any angle \( \theta \), \( \sin \theta \) must satisfy \( -1 \leq \sin \theta \leq 1 \).
Examine the given value \( \sin \theta = 3 \) and compare it to the possible range of sine values.
Since 3 is greater than 1, it lies outside the range of possible sine values.
Therefore, it is impossible for \( \sin \theta \) to equal 3 for any real angle \( \theta \).
Conclude that the statement \( \sin \theta = 3 \) is impossible.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Range of the Sine Function
The sine function outputs values only within the range of -1 to 1 for all real angles θ. This means sin θ cannot be greater than 1 or less than -1, which is crucial for determining the possibility of a given sine value.
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Domain and Range of Function Transformations
Definition of the Sine Function
Sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. Since the hypotenuse is the longest side, this ratio must always be between -1 and 1.
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5:53Graph of Sine and Cosine Function
Evaluating Trigonometric Statements
To assess whether a trigonometric statement is possible, one must check if the given value lies within the function's range and adheres to its properties. For sin θ = 3, since 3 is outside the range [-1,1], the statement is impossible.
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7:28Evaluate Composite Functions - Values Not on Unit Circle
Related Practice
Textbook Question
Determine whether each statement is possible or impossible. See Example 4. tan θ = 0.93
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Textbook Question
Perform each calculation. See Example 3. 55° 30' + 12° 44' ― 8° 15'
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Textbook Question
An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. ―5x ― 3y = 0 , x ≤ 0
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Textbook Question
Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). 82° 30'
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Textbook Question
An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. 2x + y = 0 , x ≥ 0
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