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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![**Cosine Equation and Right Triangle Explanation**
1. Cosine Equation:
\[
\cos \theta = \frac{1}{8}
\]
Given the interval for \(\theta\):
\[
\frac{3\pi}{2} < \theta < 2\pi
\]
This indicates that \(\theta\) is in the fourth quadrant, as \(\frac{3\pi}{2}\) (270°) to \(2\pi\) (360°) corresponds to this part of the unit circle.
2. Right Triangle Representation:
The diagram includes a right triangle with:
- One leg measuring \(-1\)
- Hypotenuse measuring \(8\)
This triangle visually represents the cosine value, where:
- The adjacent side in the triangle is \(-1\)
- The hypotenuse is \(8\)
The cosine function is defined as the ratio of the adjacent side to the hypotenuse, explaining why \(\cos \theta = \frac{-1}{8}\).
3. Exponential Expression:
\[
(-1)^{8/8}^2
\]
This algebraic expression showcases exponentiation.
This content, including trigonometric analysis and algebraic expression, aligns with educational material in mathematics, particularly in trigonometry and algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb805b29b-1703-42be-83a9-c544ba7b8f1b%2F7255f37e-6dff-4321-bee2-11072394e6a3%2Fqr4ljn6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Cosine Equation and Right Triangle Explanation**
1. Cosine Equation:
\[
\cos \theta = \frac{1}{8}
\]
Given the interval for \(\theta\):
\[
\frac{3\pi}{2} < \theta < 2\pi
\]
This indicates that \(\theta\) is in the fourth quadrant, as \(\frac{3\pi}{2}\) (270°) to \(2\pi\) (360°) corresponds to this part of the unit circle.
2. Right Triangle Representation:
The diagram includes a right triangle with:
- One leg measuring \(-1\)
- Hypotenuse measuring \(8\)
This triangle visually represents the cosine value, where:
- The adjacent side in the triangle is \(-1\)
- The hypotenuse is \(8\)
The cosine function is defined as the ratio of the adjacent side to the hypotenuse, explaining why \(\cos \theta = \frac{-1}{8}\).
3. Exponential Expression:
\[
(-1)^{8/8}^2
\]
This algebraic expression showcases exponentiation.
This content, including trigonometric analysis and algebraic expression, aligns with educational material in mathematics, particularly in trigonometry and algebra.
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