2π -1 Cos | =0 sin %3D 3
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![### Trigonometric Expression Simplification
**Problem Statement:**
\[ \sin^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) = \]
**Instructions:**
(Simplify your answer. Type an exact answer, using \(\pi\) as needed. Use integers or fractions for any numbers in the expression.)
**Explanation:**
The task involves simplifying the given trigonometric expression which includes the inverse sine function (also known as arcsine) and a cosine function involving \(\pi\).
1. **Start by finding \(\cos\left(\frac{2\pi}{3}\right)\):**
\(\cos\left(\frac{2\pi}{3}\right)\) is a known value on the unit circle. The angle \(\frac{2\pi}{3}\) corresponds to 120 degrees. At this angle:
\[
\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\]
2. **Next, find \(\sin^{-1}\left(-\frac{1}{2}\right)\):**
The value \( -\frac{1}{2} \) falls within the range of the inverse sine function, which is \([-1, 1]\).
The sine of which principal angle is \( -\frac{1}{2} \)? The principal value range for the arcsine function (\(\sin^{-1}\)) is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
\[
\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}
\]
**Final Answer:**
\[ \sin^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) = -\frac{\pi}{6} \]
Feel free to verify and supplement this explanation with unit circle visuals or additional examples for further clarity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b89a6c-6ca3-4f25-9a44-1409c69966fc%2F1eee4f8b-69e8-4cfd-8ce7-e4349e43cc93%2F42dvzu_processed.png&w=3840&q=75)
Transcribed Image Text:### Trigonometric Expression Simplification
**Problem Statement:**
\[ \sin^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) = \]
**Instructions:**
(Simplify your answer. Type an exact answer, using \(\pi\) as needed. Use integers or fractions for any numbers in the expression.)
**Explanation:**
The task involves simplifying the given trigonometric expression which includes the inverse sine function (also known as arcsine) and a cosine function involving \(\pi\).
1. **Start by finding \(\cos\left(\frac{2\pi}{3}\right)\):**
\(\cos\left(\frac{2\pi}{3}\right)\) is a known value on the unit circle. The angle \(\frac{2\pi}{3}\) corresponds to 120 degrees. At this angle:
\[
\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\]
2. **Next, find \(\sin^{-1}\left(-\frac{1}{2}\right)\):**
The value \( -\frac{1}{2} \) falls within the range of the inverse sine function, which is \([-1, 1]\).
The sine of which principal angle is \( -\frac{1}{2} \)? The principal value range for the arcsine function (\(\sin^{-1}\)) is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).
\[
\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}
\]
**Final Answer:**
\[ \sin^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) = -\frac{\pi}{6} \]
Feel free to verify and supplement this explanation with unit circle visuals or additional examples for further clarity.
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