3. The terminal side of an angle in standard position intersects the unit circle at point (1, -3). What is the exact value of tan (0)? 01/0 O O -√3
3. The terminal side of an angle in standard position intersects the unit circle at point (1, -3). What is the exact value of tan (0)? 01/0 O O -√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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Can someone please explain how to do this?
Ty!
![### Question 3: Trigonometric Functions and the Unit Circle
**Problem Statement:**
The terminal side of an angle \( \theta \) in standard position intersects the unit circle at the point \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \). What is the exact value of \( \tan(\theta) \)?
**Answer Choices:**
1. \( \frac{1}{2} \)
2. \( -\frac{\sqrt{3}}{2} \)
3. \( -\frac{\sqrt{3}}{3} \)
4. \( -\sqrt{3} \)
**Solution Explanation:**
To solve this problem, recall that for any point \((x, y)\) on the unit circle corresponding to an angle \( \theta \), the trigonometric functions sine, cosine, and tangent can be expressed as:
- \( \cos(\theta) = x \)
- \( \sin(\theta) = y \)
- \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \)
Given the point \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \), where:
- \( x = \frac{1}{2} \)
- \( y = -\frac{\sqrt{3}}{2} \)
We can find \( \tan(\theta) \) by dividing \( y \) by \( x \):
\[
\tan(\theta) = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}
\]
Therefore, the exact value of \( \tan(\theta) \) is \( -\sqrt{3} \).
**Correct Answer:**
- \( -\sqrt{3} \) (Option 4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd25c5f0e-549f-434f-a4b5-e32e412218b1%2Fdedd407a-fe76-46a9-a2b2-fdc97b2a7962%2Fzr04dg8_processed.png&w=3840&q=75)
Transcribed Image Text:### Question 3: Trigonometric Functions and the Unit Circle
**Problem Statement:**
The terminal side of an angle \( \theta \) in standard position intersects the unit circle at the point \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \). What is the exact value of \( \tan(\theta) \)?
**Answer Choices:**
1. \( \frac{1}{2} \)
2. \( -\frac{\sqrt{3}}{2} \)
3. \( -\frac{\sqrt{3}}{3} \)
4. \( -\sqrt{3} \)
**Solution Explanation:**
To solve this problem, recall that for any point \((x, y)\) on the unit circle corresponding to an angle \( \theta \), the trigonometric functions sine, cosine, and tangent can be expressed as:
- \( \cos(\theta) = x \)
- \( \sin(\theta) = y \)
- \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} \)
Given the point \( \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \), where:
- \( x = \frac{1}{2} \)
- \( y = -\frac{\sqrt{3}}{2} \)
We can find \( \tan(\theta) \) by dividing \( y \) by \( x \):
\[
\tan(\theta) = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}
\]
Therefore, the exact value of \( \tan(\theta) \) is \( -\sqrt{3} \).
**Correct Answer:**
- \( -\sqrt{3} \) (Option 4)
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