Textbook Question
Find the degree measure of θ if it exists. Do not use a calculator.
θ = arcsin (-√3/2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 47
Textbook Question
Find the degree measure of θ if it exists. Do not use a calculator.
θ = sin⁻¹ 2
Verified step by step guidance1
Recall the definition of the inverse sine function, \(\sin^{-1} x\), which gives the angle \(\theta\) such that \(\sin \theta = x\) and \(\theta\) lies within the range \([-90^\circ, 90^\circ]\) (or \([-\frac{\pi}{2}, \frac{\pi}{2}]\)$).
Understand the domain of the sine function: since \(\sin \theta\) can only take values between \(-1\) and \$1$ for real angles \(\theta\), the input to \(\sin^{-1}\) must be within this range.
Check the given value inside the inverse sine: here, it is \$2\(, which is outside the domain \)[-1, 1]$.
Since \$2$ is not in the domain of the inverse sine function, there is no real angle \(\theta\) such that \(\sin \theta = 2\).
Therefore, conclude that \(\theta = \sin^{-1} 2\) does not exist as a real number (no degree measure satisfies this).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of the Sine Function
The sine function outputs values only between -1 and 1 for real angles. Any input outside this range, such as 2, is not possible for sine, meaning sin⁻¹(2) does not correspond to a real angle.
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Inverse Sine Function (Arcsin)
The inverse sine function, sin⁻¹(x), returns the angle whose sine is x, but only if x is within the domain [-1, 1]. It is used to find angles from sine values, but it is undefined for values outside this range.
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Understanding Restrictions in Trigonometric Functions
Trigonometric functions have specific domains and ranges that limit their outputs and inputs. Recognizing these restrictions helps determine whether an expression like sin⁻¹(2) has a valid solution or not.
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