Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
Related questions
Question
![### Problem and Diagram Explanation
#### Problem:
In the right triangle shown below, what is the measure of angle \(S\), to the nearest minute?
#### Diagram:
- Triangle \(M A S\) is a right triangle with angle \(A\) being the right angle (90 degrees).
- Side \(M A\) is perpendicular to side \(A S\), forming the right angle.
- The hypotenuse \(M S\) is 17 units long.
- Side \(M A\) is 8 units long.
- Angle \(S\) is the angle at point \(S\), opposite side \(M A\).
To find the measure of angle \(S\) to the nearest minute, we will use trigonometry. Specifically, we can use the sine, cosine, or tangent functions to find this angle. Here, it's appropriate to use the cosine function given the known sides of the triangle.
##### Step-by-Step Solution:
1. **Using the Cosine Function:**
The cosine of angle \(S\) in a right triangle is given by the ratio of the length of the adjacent side (side \(A M\)) to the hypotenuse (side \(M S\)).
\[\cos(S) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AM}{MS}\]
Substituting in the known values:
\[
\cos(S) = \frac{8}{17}
\]
2. **Calculate the Angle:**
To find angle \(S\), take the inverse cosine (arccos) of \( \frac{8}{17} \):
\[
S = \cos^{-1} \left(\frac{8}{17}\right)
\]
3. **Converting to Degrees and Minutes:**
Using a calculator, we get:
\[
S \approx 61.927513\, \text{degrees}
\]
To convert the decimal part to minutes:
- Take the decimal part (0.927513) and multiply by 60:
\[
0.927513 \times 60 \approx 55.65078\, \text{minutes}
\]
We can round 55.65078 to the nearest minute:
\[
55.65078 \approx 56\, \text{minutes}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba437994-7eb2-41d2-bf90-86c7e632409d%2F5a671533-23bc-45d0-8b9e-fc145b305419%2Fftdm59_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem and Diagram Explanation
#### Problem:
In the right triangle shown below, what is the measure of angle \(S\), to the nearest minute?
#### Diagram:
- Triangle \(M A S\) is a right triangle with angle \(A\) being the right angle (90 degrees).
- Side \(M A\) is perpendicular to side \(A S\), forming the right angle.
- The hypotenuse \(M S\) is 17 units long.
- Side \(M A\) is 8 units long.
- Angle \(S\) is the angle at point \(S\), opposite side \(M A\).
To find the measure of angle \(S\) to the nearest minute, we will use trigonometry. Specifically, we can use the sine, cosine, or tangent functions to find this angle. Here, it's appropriate to use the cosine function given the known sides of the triangle.
##### Step-by-Step Solution:
1. **Using the Cosine Function:**
The cosine of angle \(S\) in a right triangle is given by the ratio of the length of the adjacent side (side \(A M\)) to the hypotenuse (side \(M S\)).
\[\cos(S) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AM}{MS}\]
Substituting in the known values:
\[
\cos(S) = \frac{8}{17}
\]
2. **Calculate the Angle:**
To find angle \(S\), take the inverse cosine (arccos) of \( \frac{8}{17} \):
\[
S = \cos^{-1} \left(\frac{8}{17}\right)
\]
3. **Converting to Degrees and Minutes:**
Using a calculator, we get:
\[
S \approx 61.927513\, \text{degrees}
\]
To convert the decimal part to minutes:
- Take the decimal part (0.927513) and multiply by 60:
\[
0.927513 \times 60 \approx 55.65078\, \text{minutes}
\]
We can round 55.65078 to the nearest minute:
\[
55.65078 \approx 56\, \text{minutes}
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