Find the exact value of each of the remaining trigonometric functions of 0. Rationalize denominators when applicable. √√5 cot 0 = given that is in quadrant I " 6
Find the exact value of each of the remaining trigonometric functions of 0. Rationalize denominators when applicable. √√5 cot 0 = given that is in quadrant I " 6
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 6GP
Related questions
Question
![**Task:**
Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable.
Given:
\[ \cot \theta = \frac{\sqrt{5}}{6} \]
\[ \theta \text{ is in quadrant I} \]
**Solution Overview:**
To solve this problem, we need to find the values of the remaining trigonometric functions: sin(θ), cos(θ), tan(θ), sec(θ), and csc(θ). Given that cot(θ) is in the first quadrant, we know that all trigonometric functions will be positive.
**Steps to Find the Remaining Trigonometric Functions:**
1. **Assign Variables for Trigonometric Values:**
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
From the given problem:
\[ \cot \theta = \frac{\sqrt{5}}{6} \]
2. **Assign \( \cos \theta = \sqrt{5} \) and \( \sin \theta = 6 \):**
Let's keep these values and use the Pythagorean identity to find the hypotenuse (r).
3. **Finding the Hypotenuse Using Pythagorean Theorem:**
\[ r = \sqrt{\cos^2 \theta + \sin^2 \theta} \]
Substituting \(\cos \theta \) and \(\sin \theta \):
\[ r = \sqrt{(\sqrt{5})^2 + (6)^2} = \sqrt{5 + 36} = \sqrt{41} \]
4. **Calculate the Remaining Trigonometric Values:**
- **sin(θ):**
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{\sqrt{41}} \]
- **cos(θ):**
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{5}}{\sqrt{41}} \]
- **tan(θ):**
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{6}{\sqrt{5}} \]
- **csc(θ):**
\[ \csc \theta = \frac{1}{\sin \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41fda77c-0a6e-4768-b489-0b9efc672c27%2F1af3599b-c243-4661-a5e8-b37c652a1c63%2Fe65nbj_processed.png&w=3840&q=75)
Transcribed Image Text:**Task:**
Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable.
Given:
\[ \cot \theta = \frac{\sqrt{5}}{6} \]
\[ \theta \text{ is in quadrant I} \]
**Solution Overview:**
To solve this problem, we need to find the values of the remaining trigonometric functions: sin(θ), cos(θ), tan(θ), sec(θ), and csc(θ). Given that cot(θ) is in the first quadrant, we know that all trigonometric functions will be positive.
**Steps to Find the Remaining Trigonometric Functions:**
1. **Assign Variables for Trigonometric Values:**
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
From the given problem:
\[ \cot \theta = \frac{\sqrt{5}}{6} \]
2. **Assign \( \cos \theta = \sqrt{5} \) and \( \sin \theta = 6 \):**
Let's keep these values and use the Pythagorean identity to find the hypotenuse (r).
3. **Finding the Hypotenuse Using Pythagorean Theorem:**
\[ r = \sqrt{\cos^2 \theta + \sin^2 \theta} \]
Substituting \(\cos \theta \) and \(\sin \theta \):
\[ r = \sqrt{(\sqrt{5})^2 + (6)^2} = \sqrt{5 + 36} = \sqrt{41} \]
4. **Calculate the Remaining Trigonometric Values:**
- **sin(θ):**
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{\sqrt{41}} \]
- **cos(θ):**
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{5}}{\sqrt{41}} \]
- **tan(θ):**
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{6}{\sqrt{5}} \]
- **csc(θ):**
\[ \csc \theta = \frac{1}{\sin \
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,