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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![### Problem Statement:
- **We have**
\[ \tan^3 \theta = 3 \]
- **What’s the value of** \( \theta \)?
- **Explain in detail**.
### Solution:
To solve for \( \theta \) given \( \tan^3 \theta = 3 \), follow these steps:
1. **Understand the problem**: We need to find the value of \( \theta \) such that when the tangent of \( \theta \) is raised to the power of 3, it equals 3.
2. **Isolate the tangent function**:
\[
\tan^3 \theta = 3
\]
Taking the cube root of both sides, we get:
\[
\tan \theta = \sqrt[3]{3}
\]
3. **Find \( \theta \)**:
\[
\theta = \tan^{-1}(\sqrt[3]{3})
\]
4. **Use a calculator or reference table** to find the numerical value of \( \theta \):
\[
\theta \approx \tan^{-1}(1.4422) \approx 55^\circ
\]
### Detailed Explanation:
- **Step 1**: Recognize that the equation involves the tangent function raised to a power. The \( \theta \) we are looking for must satisfy this trigonometric identity.
- **Step 2**: Isolating the tangent function allows us to simplify the problem.
- **Step 3**: Inverse tangent (or arctan) is used to find the angle whose tangent is \( \sqrt[3]{3} \).
- **Step 4**: Approximate this value using a calculator for practical purposes. Make sure your calculator is in the correct mode (degrees or radians) based on the context of the problem.
By following these steps, we determine that \( \theta \approx 55^\circ \) for \( \tan^3 \theta = 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2Fddafe3b2-b5c4-4dcc-981d-611f06d3d155%2Fh330upl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
- **We have**
\[ \tan^3 \theta = 3 \]
- **What’s the value of** \( \theta \)?
- **Explain in detail**.
### Solution:
To solve for \( \theta \) given \( \tan^3 \theta = 3 \), follow these steps:
1. **Understand the problem**: We need to find the value of \( \theta \) such that when the tangent of \( \theta \) is raised to the power of 3, it equals 3.
2. **Isolate the tangent function**:
\[
\tan^3 \theta = 3
\]
Taking the cube root of both sides, we get:
\[
\tan \theta = \sqrt[3]{3}
\]
3. **Find \( \theta \)**:
\[
\theta = \tan^{-1}(\sqrt[3]{3})
\]
4. **Use a calculator or reference table** to find the numerical value of \( \theta \):
\[
\theta \approx \tan^{-1}(1.4422) \approx 55^\circ
\]
### Detailed Explanation:
- **Step 1**: Recognize that the equation involves the tangent function raised to a power. The \( \theta \) we are looking for must satisfy this trigonometric identity.
- **Step 2**: Isolating the tangent function allows us to simplify the problem.
- **Step 3**: Inverse tangent (or arctan) is used to find the angle whose tangent is \( \sqrt[3]{3} \).
- **Step 4**: Approximate this value using a calculator for practical purposes. Make sure your calculator is in the correct mode (degrees or radians) based on the context of the problem.
By following these steps, we determine that \( \theta \approx 55^\circ \) for \( \tan^3 \theta = 3 \).
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