Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = cos⁻¹ (―1)
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 23
Textbook Question
Find the exact value of each real number y if it exists. Do not use a calculator.
y = arccos (―√3/2)
Verified step by step guidance1
Recall that the function \( y = \arccos(x) \) gives the angle \( y \) in the range \( [0, \pi] \) such that \( \cos(y) = x \). Here, we need to find \( y \) such that \( \cos(y) = -\frac{\sqrt{3}}{2} \).
Identify the reference angle whose cosine value is \( \frac{\sqrt{3}}{2} \). From common trigonometric values, \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \).
Since the cosine value is negative, determine in which quadrants cosine is negative. Cosine is negative in the second and third quadrants, but \( \arccos \) only returns values in \( [0, \pi] \), which corresponds to the first and second quadrants.
Find the angle in the second quadrant with the same reference angle \( \frac{\pi}{6} \). This angle is \( \pi - \frac{\pi}{6} = \frac{5\pi}{6} \).
Therefore, the exact value of \( y = \arccos\left(-\frac{\sqrt{3}}{2}\right) \) is \( \frac{5\pi}{6} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, like arccos, return the angle whose trigonometric ratio equals a given value. For arccos(x), it gives the angle y in [0, π] such that cos(y) = x. Understanding their domain and range is essential to find exact angle values.
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Introduction to Inverse Trig Functions
Exact Values of Cosine for Special Angles
Certain angles have well-known cosine values expressed in terms of square roots, such as cos(π/6) = √3/2 and cos(5π/6) = -√3/2. Recognizing these exact values helps in identifying the angle corresponding to a given cosine value without a calculator.
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Range and Principal Values of arccos
The arccos function outputs principal values in the interval [0, π]. When given a negative cosine value like -√3/2, the corresponding angle must lie within this range, typically in the second quadrant, ensuring the solution is unique and exact.
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