Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Find \(\sin(\theta)\) and \(\cos(\theta)\). Give exact answers.**
### Diagram Explanation:
The image shows a right triangle with:
- One angle labeled \(\theta\),
- The opposite side of \(\theta\) is labeled with a length of 32,
- The adjacent side to \(\theta\) is labeled with a length of 12.
### Solution:
To find \(\sin(\theta)\) and \(\cos(\theta)\), we first use the Pythagorean theorem to determine the hypotenuse.
The Pythagorean theorem states:
\[
c^2 = a^2 + b^2
\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
Here, \(a = 32\) and \(b = 12\):
\[
c^2 = 32^2 + 12^2 = 1024 + 144 = 1168
\]
Thus,
\[
c = \sqrt{1168} = \sqrt{4 \times 292} = 2\sqrt{292} = 2\sqrt{4 \times 73} = 4\sqrt{73}
\]
### Trigonometric Functions:
- **Sine Function**: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\[
\sin(\theta) = \frac{32}{4\sqrt{73}} = \frac{8}{\sqrt{73}}
\]
- **Cosine Function**: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\[
\cos(\theta) = \frac{12}{4\sqrt{73}} = \frac{3}{\sqrt{73}}
\]
For exact answers, \(\sin(\theta) = \frac{8}{\sqrt{73}}\) and \(\cos(\theta) = \frac{3}{\sqrt{73}}\). To rationalize, multiply numerator and denominator by \(\sqrt{73}\):
- \(\sin(\theta) = \frac{8\sqrt{73}}{73}\)
- \(\cos(\theta) = \frac{3\sqrt{73}}{73](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd627bc06-52c0-4d11-87e8-f3ff50368dcf%2F216a5cf0-b09a-4a4d-afa9-a1d71488d6cf%2Fcqyrwtm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find \(\sin(\theta)\) and \(\cos(\theta)\). Give exact answers.**
### Diagram Explanation:
The image shows a right triangle with:
- One angle labeled \(\theta\),
- The opposite side of \(\theta\) is labeled with a length of 32,
- The adjacent side to \(\theta\) is labeled with a length of 12.
### Solution:
To find \(\sin(\theta)\) and \(\cos(\theta)\), we first use the Pythagorean theorem to determine the hypotenuse.
The Pythagorean theorem states:
\[
c^2 = a^2 + b^2
\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
Here, \(a = 32\) and \(b = 12\):
\[
c^2 = 32^2 + 12^2 = 1024 + 144 = 1168
\]
Thus,
\[
c = \sqrt{1168} = \sqrt{4 \times 292} = 2\sqrt{292} = 2\sqrt{4 \times 73} = 4\sqrt{73}
\]
### Trigonometric Functions:
- **Sine Function**: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
\[
\sin(\theta) = \frac{32}{4\sqrt{73}} = \frac{8}{\sqrt{73}}
\]
- **Cosine Function**: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
\[
\cos(\theta) = \frac{12}{4\sqrt{73}} = \frac{3}{\sqrt{73}}
\]
For exact answers, \(\sin(\theta) = \frac{8}{\sqrt{73}}\) and \(\cos(\theta) = \frac{3}{\sqrt{73}}\). To rationalize, multiply numerator and denominator by \(\sqrt{73}\):
- \(\sin(\theta) = \frac{8\sqrt{73}}{73}\)
- \(\cos(\theta) = \frac{3\sqrt{73}}{73
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 3 steps with 1 images
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education