Textbook Question
In Exercises 1–26, find the exact value of each expression. _ sec⁻¹ (−√2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
2:45 minutesProblem 32
Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ sin⁻¹ (− √3/2)
Verified step by step guidance1
Recognize that the expression is asking for the angle \( \theta \) such that \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) and \( \theta \) is in the range of the inverse sine function, which is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
Recall the reference angle where \( \sin(\theta) = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \) because \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).
Since the sine value is negative, and the inverse sine function outputs angles only between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), the angle must be in the fourth quadrant (negative angle).
Therefore, the angle \( \theta \) is the negative of the reference angle: \( \theta = -\frac{\pi}{3} \).
Conclude that \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \) as the exact value.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function (sin⁻¹ or arcsin)
The inverse sine function, denoted sin⁻¹ or arcsin, returns the angle whose sine is a given value. Its output range is limited to angles between -π/2 and π/2 (or -90° to 90°) to ensure it is a function. Understanding this range is crucial for finding the correct angle without a calculator.
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Exact Values of Sine for Special Angles
Certain angles have well-known sine values expressed in terms of square roots, such as sin(π/3) = √3/2. Recognizing these exact values allows you to identify the angle corresponding to a given sine value without approximation or a calculator.
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Sign and Quadrant Considerations for Inverse Trigonometric Functions
Since sin⁻¹ returns angles only in the first and fourth quadrants, the sign of the sine value determines the angle's sign. For negative sine values, the angle lies between -π/2 and 0. This helps in correctly assigning the angle's sign and value when evaluating inverse sine expressions.
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