Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = arcsin (―√3/2)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 29
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = csc⁻¹ (―2)
Verified step by step guidance1
Recall that the function \( y = \csc^{-1}(x) \) is the inverse cosecant function, which gives the angle \( y \) such that \( \csc y = x \). Here, we want to find \( y \) such that \( \csc y = -2 \).
Rewrite the equation \( \csc y = -2 \) in terms of sine, since \( \csc y = \frac{1}{\sin y} \). This gives \( \frac{1}{\sin y} = -2 \), so \( \sin y = -\frac{1}{2} \).
Determine the general solutions for \( y \) where \( \sin y = -\frac{1}{2} \). Recall that sine is negative in the third and fourth quadrants, and the reference angle for \( \sin y = \frac{1}{2} \) is \( \frac{\pi}{6} \).
Write the specific solutions for \( y \) in the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) or the principal range of \( \csc^{-1} \), which is usually \( [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \), excluding zero. Identify which of these solutions fall into the principal range.
Express the exact value(s) of \( y \) in radians that satisfy the equation \( y = \csc^{-1}(-2) \) within the principal range, using the reference angle \( \frac{\pi}{6} \) and the appropriate sign.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Cosecant Function (csc⁻¹)
The inverse cosecant function, csc⁻¹(x), returns the angle whose cosecant is x. It is defined for |x| ≥ 1, and its output is typically restricted to angles in the ranges [-π/2, 0) ∪ (0, π/2] to ensure it is a function.
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Graphs of Secant and Cosecant Functions
Relationship Between Cosecant and Sine
Cosecant is the reciprocal of sine, so csc(θ) = 1/sin(θ). To find y = csc⁻¹(-2), we look for an angle y where sin(y) = -1/2, which helps in determining the exact angle without a calculator.
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Exact Values of Sine for Special Angles
Certain angles have well-known sine values, such as sin(π/6) = 1/2. Using these, we can find angles where sine equals -1/2, specifically in the third and fourth quadrants, to determine the exact value of y.
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