Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ [tan(− π/6)]
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
1:58 minutesProblem 51
Textbook Question
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. sin⁻¹ (sin π)
Verified step by step guidance1
Recall that the function \(\sin^{-1}(x)\), also known as arcsine, is the inverse of the sine function but its output (range) is restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).
Identify the input to the arcsine function: here it is \(\sin \pi\). Since \(\sin \pi = 0\), rewrite the expression as \(\sin^{-1}(0)\).
Now, find the angle \(\theta\) within the range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) such that \(\sin \theta = 0\).
Recall that \(\sin \theta = 0\) at \(\theta = 0\), \(\pi\), \(2\pi\), etc., but only \(\theta = 0\) lies within the principal range of arcsine.
Therefore, the exact value of \(\sin^{-1}(\sin \pi)\) is the angle \(\theta = 0\).
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Video duration:
1mKey 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 is restricted to the principal range of [-π/2, π/2] to ensure it is a proper function. Understanding this range is crucial when evaluating expressions like sin⁻¹(sin θ).
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Sine Function Periodicity and Symmetry
The sine function is periodic with period 2π, meaning sin(θ) = sin(θ + 2πk) for any integer k. It is also symmetric about the origin (odd function). These properties help simplify expressions and find equivalent angles within the principal range of the inverse sine.
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Evaluating sin⁻¹(sin θ) for Angles Outside the Principal Range
When θ is outside the principal range of arcsin, sin⁻¹(sin θ) equals the angle within [-π/2, π/2] that has the same sine value as θ. This often involves finding a reference angle or using angle identities to map θ back into the principal range.
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