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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 (MindTap Course List)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section: Chapter Questions
Problem 1RP: The origins of the sine function are found in the tables of chords for a circle constructed by the...
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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)
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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