**Question:** Find the value of cos θ for the triangle shown below. **Diagram:** The diagram shows a right-angled triangle with: - Adjacent side labeled as 1 - Opposite side labeled as 2 - Hypotenuse labeled as √5 **Answer Choices:** 1. \(\frac{\sqrt{5}}{2}\) 2. \(\frac{\sqrt{5}}{5}\) 3. \(\sqrt{5}\) 4. \(\frac{2}{\sqrt{5}}\) **Explanation:** To find the cosine of angle θ (cos θ), use the definition of cosine in a right-angled triangle which is the ratio of the length of the adjacent side to the length of the hypotenuse. Mathematically, \[ \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \] By substituting the given lengths, \[ \cos θ = \frac{1}{\sqrt{5}} \] However, it is often useful to rationalize the denominator: \[ \cos θ = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \] Thus, the value of cos θ is \(\frac{\sqrt{5}}{5}\), which corresponds to the second answer choice.
**Question:** Find the value of cos θ for the triangle shown below. **Diagram:** The diagram shows a right-angled triangle with: - Adjacent side labeled as 1 - Opposite side labeled as 2 - Hypotenuse labeled as √5 **Answer Choices:** 1. \(\frac{\sqrt{5}}{2}\) 2. \(\frac{\sqrt{5}}{5}\) 3. \(\sqrt{5}\) 4. \(\frac{2}{\sqrt{5}}\) **Explanation:** To find the cosine of angle θ (cos θ), use the definition of cosine in a right-angled triangle which is the ratio of the length of the adjacent side to the length of the hypotenuse. Mathematically, \[ \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \] By substituting the given lengths, \[ \cos θ = \frac{1}{\sqrt{5}} \] However, it is often useful to rationalize the denominator: \[ \cos θ = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \] Thus, the value of cos θ is \(\frac{\sqrt{5}}{5}\), which corresponds to the second answer choice.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Question:**
Find the value of cos θ for the triangle shown below.
**Diagram:**
The diagram shows a right-angled triangle with:
- Adjacent side labeled as 1
- Opposite side labeled as 2
- Hypotenuse labeled as √5
**Answer Choices:**
1. \(\frac{\sqrt{5}}{2}\)
2. \(\frac{\sqrt{5}}{5}\)
3. \(\sqrt{5}\)
4. \(\frac{2}{\sqrt{5}}\)
**Explanation:**
To find the cosine of angle θ (cos θ), use the definition of cosine in a right-angled triangle which is the ratio of the length of the adjacent side to the length of the hypotenuse.
Mathematically,
\[ \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \]
By substituting the given lengths,
\[ \cos θ = \frac{1}{\sqrt{5}} \]
However, it is often useful to rationalize the denominator:
\[ \cos θ = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \]
Thus, the value of cos θ is \(\frac{\sqrt{5}}{5}\), which corresponds to the second answer choice.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1af3b856-d446-4ff7-a63e-89e70c8da553%2Ff05d57c8-da27-4914-a43d-16da1382d6f5%2Fcsiwh8.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Find the value of cos θ for the triangle shown below.
**Diagram:**
The diagram shows a right-angled triangle with:
- Adjacent side labeled as 1
- Opposite side labeled as 2
- Hypotenuse labeled as √5
**Answer Choices:**
1. \(\frac{\sqrt{5}}{2}\)
2. \(\frac{\sqrt{5}}{5}\)
3. \(\sqrt{5}\)
4. \(\frac{2}{\sqrt{5}}\)
**Explanation:**
To find the cosine of angle θ (cos θ), use the definition of cosine in a right-angled triangle which is the ratio of the length of the adjacent side to the length of the hypotenuse.
Mathematically,
\[ \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \]
By substituting the given lengths,
\[ \cos θ = \frac{1}{\sqrt{5}} \]
However, it is often useful to rationalize the denominator:
\[ \cos θ = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \]
Thus, the value of cos θ is \(\frac{\sqrt{5}}{5}\), which corresponds to the second answer choice.
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