Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.1: Angles
Problem 3E
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Question
![**Problem Statement**
*Find the area of the shaded circle segment to the nearest tenth.*
**Image Description and Explanation**
The image depicts a circle with a radius of 1 centimeter, marked with points S, T, and R. Point R is the center of the circle, and points S and T lie on the circumference. Triangle SRT is a right triangle with an angle of 90° at point R.
**Diagram Explanation**
- **Circle**: The circle has a radius (R) of 1 cm.
- **Right Triangle SRT**: This triangle is inscribed within the circle. The right angle (90°) is at point R.
- **Sector and Segment**: The shaded region represents the segment of the circle formed by the chord ST and the arc ST.
**Calculation Steps**
1. **Area of the sector (Sector SRT)**:
The area of a sector with a central angle of 90° (or π/2 radians) can be calculated using the formula:
\[
\text{Area of Sector} = \frac{\theta}{2\pi} \times \pi R^2
\]
Where:
- \(\theta = \frac{\pi}{2}\) radians (90°),
- \(R = 1\) cm.
2. **Area of Triangle SRT**:
Since SRT is a right triangle, its area can be calculated using:
\[
\text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Here, both base and height are equal to the radius (1 cm).
3. **Area of the Shaded Segment**:
The area of the shaded segment is found by subtracting the area of the triangle from the area of the sector:
\[
\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}
\]
**Answer Submission**
The answer should be written in the provided text box as:
\[ \text{Answer: } \_\_\_\_ \, \text{cm}^2 \]
**Note to Students**
Ensure to round your final answer to the nearest tenth as instructed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03ee4b44-fd9b-4d43-bc23-25b15a4cdf81%2Fb4bc5178-6e2b-4508-86d0-3a9959fdd78c%2Fyx5sj7f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
*Find the area of the shaded circle segment to the nearest tenth.*
**Image Description and Explanation**
The image depicts a circle with a radius of 1 centimeter, marked with points S, T, and R. Point R is the center of the circle, and points S and T lie on the circumference. Triangle SRT is a right triangle with an angle of 90° at point R.
**Diagram Explanation**
- **Circle**: The circle has a radius (R) of 1 cm.
- **Right Triangle SRT**: This triangle is inscribed within the circle. The right angle (90°) is at point R.
- **Sector and Segment**: The shaded region represents the segment of the circle formed by the chord ST and the arc ST.
**Calculation Steps**
1. **Area of the sector (Sector SRT)**:
The area of a sector with a central angle of 90° (or π/2 radians) can be calculated using the formula:
\[
\text{Area of Sector} = \frac{\theta}{2\pi} \times \pi R^2
\]
Where:
- \(\theta = \frac{\pi}{2}\) radians (90°),
- \(R = 1\) cm.
2. **Area of Triangle SRT**:
Since SRT is a right triangle, its area can be calculated using:
\[
\text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}
\]
Here, both base and height are equal to the radius (1 cm).
3. **Area of the Shaded Segment**:
The area of the shaded segment is found by subtracting the area of the triangle from the area of the sector:
\[
\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}
\]
**Answer Submission**
The answer should be written in the provided text box as:
\[ \text{Answer: } \_\_\_\_ \, \text{cm}^2 \]
**Note to Students**
Ensure to round your final answer to the nearest tenth as instructed.
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