Textbook Question
In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. _ cos⁻¹ √5/7
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
2:26 minutesProblem 39
Textbook Question
In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ cos(sin⁻¹ √2/2)
Verified step by step guidance1
Recognize that the expression is \( \cos(\sin^{-1}(\frac{\sqrt{2}}{2})) \). Here, \( \sin^{-1} \) is the inverse sine function, which gives an angle \( \theta \) such that \( \sin \theta = \frac{\sqrt{2}}{2} \).
Set \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \), so by definition, \( \sin \theta = \frac{\sqrt{2}}{2} \). Our goal is to find \( \cos \theta \).
Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \frac{\sqrt{2}}{2} \) into this identity to find \( \cos \theta \).
Calculate \( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left( \frac{\sqrt{2}}{2} \right)^2 \). Simplify the right side to find \( \cos^2 \theta \).
Take the square root of \( \cos^2 \theta \) to find \( \cos \theta \). Consider the range of \( \theta = \sin^{-1}(\frac{\sqrt{2}}{2}) \) to determine the correct sign (positive or negative) of \( \cos \theta \).
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Video duration:
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, sin⁻¹(x), returns the angle whose sine is x. It is defined for x in the interval [-1, 1] and outputs angles in the range [-π/2, π/2]. Understanding this helps identify the angle corresponding to a given sine value.
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Relationship Between Sine and Cosine
Sine and cosine are co-functions related by the Pythagorean identity: sin²θ + cos²θ = 1. Knowing one trigonometric value allows you to find the other by rearranging this identity, which is essential for evaluating expressions like cos(sin⁻¹ x).
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Exact Values of Special Angles
Certain angles have well-known exact sine and cosine values, such as π/4 (45°), where sin(π/4) = cos(π/4) = √2/2. Recognizing these special angles helps in finding exact trigonometric values without a calculator.
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