Textbook Question
Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 13
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = sin⁻¹ 0
Verified step by step guidance1
Understand that the problem asks for the values of \(y\) such that \(\sin y = 0\), where \(y = \sin^{-1} 0\) represents the inverse sine (arcsine) function.
Recall the definition of the inverse sine function: \(y = \sin^{-1} x\) means \(\sin y = x\) and \(y\) lies within the principal range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
Set up the equation based on the problem: \(\sin y = 0\) with \(y \in [-\frac{\pi}{2}, \frac{\pi}{2}]\).
Identify all angles \(y\) in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) where \(\sin y = 0\). These are the values where the sine function crosses zero within the principal range.
Conclude that the exact value(s) of \(y\) satisfying the equation are those angles found in the previous step, which represent the output of \(\sin^{-1} 0\).
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Key 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 as sin⁻¹ or arcsin, returns the angle whose sine is a given number. It is defined for inputs between -1 and 1 and outputs angles in the range [-π/2, π/2]. Understanding this function helps find angles corresponding to specific sine values.
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Sine Function Values
The sine function relates an angle to the ratio of the opposite side over the hypotenuse in a right triangle. Key values include sin(0) = 0, sin(π/2) = 1, and sin(-π/2) = -1. Recognizing these standard values aids in identifying angles for given sine outputs.
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Domain and Range Restrictions of Inverse Trigonometric Functions
Inverse trigonometric functions have restricted ranges to ensure they are functions. For arcsin, the output is limited to angles between -π/2 and π/2. This restriction means that for sin⁻¹(0), the principal value is 0, even though sine of other angles like π also equals zero.
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