Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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Question
![**Mathematical Problem Breakdown**
**Topic:** Using Addition or Subtraction Formula to Simplify Trigonometric Expressions
**Objective:** Learn how to use an addition or subtraction formula in trigonometry to find the exact value of a given expression and rationalize the denominator if necessary.
---
### Problem Statement
Given the trigonometric expression:
\[
\frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
### Instructions:
1. **Use an addition or subtraction formula** to find the exact value of the given expression in simplest form.
2. **Rationalize your denominator**, if necessary.
#### Explanation of the Problem
- **Trigonometric Functions:** This problem involves the tangent function, which is one of the primary trigonometric functions.
- **Angles Given:** The angles provided in the problem are 15 degrees and 45 degrees.
### Using the Tangent Subtraction Formula
The tangent subtraction formula is given by:
\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\]
Here, \(A = 15^\circ\) and \(B = 45^\circ\).
Plugging the values into the formula:
\[
\tan(15^\circ - 45^\circ) = \frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
\[
\tan(-30^\circ) = \frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
### Simplifying the Expression
- We know that \(\tan(-\theta) = -\tan(\theta)\).
- Also, \(\tan 45^\circ = 1\).
So,
\[
\tan(-30^\circ) = -\tan 30^\circ
\]
Using the known value, \(\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\):
\[
\tan(-30^\circ) = -\frac{\sqrt{3}}{3}
\]
### Final Answer
\[
\frac{\tan 15^\circ - \tan 45^\](https://content.bartleby.com/qna-images/question/e92e694a-8dce-4f5c-a0b3-7db66cedb1e5/1500718c-89b2-42be-8eae-78a8b33bf26e/o4x8t4h_processed.jpeg)
Transcribed Image Text:**Mathematical Problem Breakdown**
**Topic:** Using Addition or Subtraction Formula to Simplify Trigonometric Expressions
**Objective:** Learn how to use an addition or subtraction formula in trigonometry to find the exact value of a given expression and rationalize the denominator if necessary.
---
### Problem Statement
Given the trigonometric expression:
\[
\frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
### Instructions:
1. **Use an addition or subtraction formula** to find the exact value of the given expression in simplest form.
2. **Rationalize your denominator**, if necessary.
#### Explanation of the Problem
- **Trigonometric Functions:** This problem involves the tangent function, which is one of the primary trigonometric functions.
- **Angles Given:** The angles provided in the problem are 15 degrees and 45 degrees.
### Using the Tangent Subtraction Formula
The tangent subtraction formula is given by:
\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\]
Here, \(A = 15^\circ\) and \(B = 45^\circ\).
Plugging the values into the formula:
\[
\tan(15^\circ - 45^\circ) = \frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
\[
\tan(-30^\circ) = \frac{\tan 15^\circ - \tan 45^\circ}{1 + \tan 15^\circ \tan 45^\circ}
\]
### Simplifying the Expression
- We know that \(\tan(-\theta) = -\tan(\theta)\).
- Also, \(\tan 45^\circ = 1\).
So,
\[
\tan(-30^\circ) = -\tan 30^\circ
\]
Using the known value, \(\tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\):
\[
\tan(-30^\circ) = -\frac{\sqrt{3}}{3}
\]
### Final Answer
\[
\frac{\tan 15^\circ - \tan 45^\
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