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Use the half-angle formula for cosine to compute cos() given cos(0) = 47 50 (Leave your answer in exact form.) where 0 < 0 <

Use the half-angle formula for cosine to compute cos() given cos(0) = 47 50 (Leave your answer in exact form.) where 0 < 0 <

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 81E
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### Half-Angle Formula for Cosine **Problem Statement:** Use the half-angle formula for cosine to compute \(\cos\left(\frac{\theta}{2}\right)\) given \(\cos(\theta) = \frac{47}{50}\) where \(0 < \theta < \frac{\pi}{2}\). **Instructions:** - Leave your answer in exact form. **Explanation:** The half-angle formula for cosine is applied to determine the value of \( \cos\left(\frac{\theta}{2}\right) \). The given problem provides information including the value of \(\cos(\theta)\) and the range within which \(\theta\) lies. **Formula:** \[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \] Since \(0 < \theta < \frac{\pi}{2}\), the half-angle \(\frac{\theta}{2}\) is also within the first quadrant, meaning \( \cos\left(\frac{\theta}{2}\right) \) is non-negative. Thus the correct form of the formula in this context is: \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \] Substitute the given value of \(\cos(\theta)\): \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}} \] **Final Answer:** \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}} \] The answer should be left in this exact form as requested.
Transcribed Image Text:### Half-Angle Formula for Cosine **Problem Statement:** Use the half-angle formula for cosine to compute \(\cos\left(\frac{\theta}{2}\right)\) given \(\cos(\theta) = \frac{47}{50}\) where \(0 < \theta < \frac{\pi}{2}\). **Instructions:** - Leave your answer in exact form. **Explanation:** The half-angle formula for cosine is applied to determine the value of \( \cos\left(\frac{\theta}{2}\right) \). The given problem provides information including the value of \(\cos(\theta)\) and the range within which \(\theta\) lies. **Formula:** \[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \] Since \(0 < \theta < \frac{\pi}{2}\), the half-angle \(\frac{\theta}{2}\) is also within the first quadrant, meaning \( \cos\left(\frac{\theta}{2}\right) \) is non-negative. Thus the correct form of the formula in this context is: \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \] Substitute the given value of \(\cos(\theta)\): \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}} \] **Final Answer:** \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{47}{50}}{2}} \] The answer should be left in this exact form as requested.
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