What is sec(¹3)? -√[?] घ
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°.
Related questions
Question
Hey what’s the answer to this
![**Title: Understanding Trigonometric Functions: Example Problem**
**Question:**
What is sec(13π/4)?
**Solution:**
\[ \sec\left(\frac{13\pi}{4}\right) \]
**Answer:**
\[ -\sqrt{\boxed{2}} \]
**Explanation:**
To solve this problem, we need to calculate the secant of \( \frac{13\pi}{4} \).
1. **Understanding the Angle:**
- First, reduce the angle \( \frac{13\pi}{4} \) by converting it into an equivalent angle within the principal interval \([0, 2\pi)\).
- \( \frac{13\pi}{4} \) can be written as \( 3\pi + \frac{\pi}{4} \).
2. **Simplification:**
- Since \(3\pi\) corresponds to \(1.5 \times (2\pi)\), it's equivalent to \( \pi \).
- Hence, \( \frac{13\pi}{4} = \pi + \frac{\pi}{4} \).
3. **New Equivalent Angle:**
- The equivalent angle is \( \frac{\pi}{4} \).
4. **Evaluating Secant Function:**
- Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
- For \( \theta = \frac{\pi}{4} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
5. **Calculation:**
- \( \sec(\frac{\pi}{4}) = \frac{1}{\cos(\frac{\pi}{4})} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \).
6. **Checking the Sign:**
- Because \(3\pi\) adds a negative sign (cosine function behavior in the third quadrant), the result is \( -\sqrt{2} \).
So, \( \sec\left(\frac{13\pi}{4}\right) = -\sqrt{2} \).
Here, the final answer is represented as:
\[ -\sqrt{\boxed{2}} \]
**Note:**
The question and the process explained are useful for students to understand how to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88948349-7a3a-42b1-b43e-0f6310456bad%2F0ab2f2f4-aadd-4670-a548-d233de9691ac%2Fghbfc4f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Trigonometric Functions: Example Problem**
**Question:**
What is sec(13π/4)?
**Solution:**
\[ \sec\left(\frac{13\pi}{4}\right) \]
**Answer:**
\[ -\sqrt{\boxed{2}} \]
**Explanation:**
To solve this problem, we need to calculate the secant of \( \frac{13\pi}{4} \).
1. **Understanding the Angle:**
- First, reduce the angle \( \frac{13\pi}{4} \) by converting it into an equivalent angle within the principal interval \([0, 2\pi)\).
- \( \frac{13\pi}{4} \) can be written as \( 3\pi + \frac{\pi}{4} \).
2. **Simplification:**
- Since \(3\pi\) corresponds to \(1.5 \times (2\pi)\), it's equivalent to \( \pi \).
- Hence, \( \frac{13\pi}{4} = \pi + \frac{\pi}{4} \).
3. **New Equivalent Angle:**
- The equivalent angle is \( \frac{\pi}{4} \).
4. **Evaluating Secant Function:**
- Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
- For \( \theta = \frac{\pi}{4} \), \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
5. **Calculation:**
- \( \sec(\frac{\pi}{4}) = \frac{1}{\cos(\frac{\pi}{4})} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \).
6. **Checking the Sign:**
- Because \(3\pi\) adds a negative sign (cosine function behavior in the third quadrant), the result is \( -\sqrt{2} \).
So, \( \sec\left(\frac{13\pi}{4}\right) = -\sqrt{2} \).
Here, the final answer is represented as:
\[ -\sqrt{\boxed{2}} \]
**Note:**
The question and the process explained are useful for students to understand how to
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