Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Could you help me solve this equation:
"Draw a right triangle to simplify
the given expressions. Assume x > 0."
![**Transcription:**
63. \( \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \)
**Explanation:**
This mathematical expression involves the sine and arccosine (inverse cosine) functions. Here, \( \cos^{-1} \left( \frac{x}{2} \right) \) represents the angle whose cosine is \( \frac{x}{2} \). The sine of this angle is then taken, resulting in the overall expression \( \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \).
In terms of trigonometry, this can also be understood using the Pythagorean identity. If \( \theta = \cos^{-1}\left( \frac{x}{2} \right) \), then:
\[
\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left( \frac{x}{2} \right)^2}
\]
This expression calculates the sine of the angle whose cosine is \( \frac{x}{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f38dbf2-543e-48ee-b7bf-c08fbbbcda3f%2Fbeaf976a-abb8-4f45-8c39-31fc27beae12%2Fv1otina_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription:**
63. \( \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \)
**Explanation:**
This mathematical expression involves the sine and arccosine (inverse cosine) functions. Here, \( \cos^{-1} \left( \frac{x}{2} \right) \) represents the angle whose cosine is \( \frac{x}{2} \). The sine of this angle is then taken, resulting in the overall expression \( \sin \left( \cos^{-1} \left( \frac{x}{2} \right) \right) \).
In terms of trigonometry, this can also be understood using the Pythagorean identity. If \( \theta = \cos^{-1}\left( \frac{x}{2} \right) \), then:
\[
\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left( \frac{x}{2} \right)^2}
\]
This expression calculates the sine of the angle whose cosine is \( \frac{x}{2} \).
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