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Evaluate cos (sin + cos 2

Evaluate cos (sin + cos 2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 28RE
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**Instruction for Students:** Rationalize denominators when applicable, DO NOT give any calculator value!!!
Transcribed Image Text:**Instruction for Students:** Rationalize denominators when applicable, DO NOT give any calculator value!!!
**Problem 8: Evaluate the following expression** \[ \cos \left( \sin^{-1} \frac{1}{2} + \cos^{-1} \frac{1}{2} \right) \] **Solution:** To solve this expression, we use the properties of inverse trigonometric functions: 1. Let \( x = \sin^{-1} \frac{1}{2} \). 2. Let \( y = \cos^{-1} \frac{1}{2} \). We know that: \[ \sin x = \frac{1}{2} \quad \Rightarrow \quad x = \frac{\pi}{6} \] and \[ \cos y = \frac{1}{2} \quad \Rightarrow \quad y = \frac{\pi}{3} \] Thus, our expression becomes: \[ \cos \left( x + y \right) = \cos \left( \frac{\pi}{6} + \frac{\pi}{3} \right) \] Adding the angles: \[ \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \] Then: \[ \cos \left( \frac{\pi}{2} \right) = 0 \] Thus, the value is: \[ \boxed{0} \]
Transcribed Image Text:**Problem 8: Evaluate the following expression** \[ \cos \left( \sin^{-1} \frac{1}{2} + \cos^{-1} \frac{1}{2} \right) \] **Solution:** To solve this expression, we use the properties of inverse trigonometric functions: 1. Let \( x = \sin^{-1} \frac{1}{2} \). 2. Let \( y = \cos^{-1} \frac{1}{2} \). We know that: \[ \sin x = \frac{1}{2} \quad \Rightarrow \quad x = \frac{\pi}{6} \] and \[ \cos y = \frac{1}{2} \quad \Rightarrow \quad y = \frac{\pi}{3} \] Thus, our expression becomes: \[ \cos \left( x + y \right) = \cos \left( \frac{\pi}{6} + \frac{\pi}{3} \right) \] Adding the angles: \[ \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \] Then: \[ \cos \left( \frac{\pi}{2} \right) = 0 \] Thus, the value is: \[ \boxed{0} \]
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