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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### 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.
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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