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
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Question
![**Mathematics Problem: Right Triangle Angle Calculation**
**Problem Statement:**
In the diagram below of right triangle \(KTW\), \(KW = 6\), \(KT = 5\), and \(m\angle KTW = 90^\circ\).
\[
\begin{array}{ccccc}
&&W & \\
&&| & \\
&&| &6 \\
T& \underline{\blacksquare} &\hspace{0.5cm} & \_ \_ \_ \_ \_ \_ &_K\,5
\end{array}
\]
**Question:**
What is the measure of \( \angle K \), to the nearest minute?
### Explanation of Diagram:
The diagram represents a right-angled triangle \(KTW\) where:
- \(W\) is the vertex opposite the right angle.
- \( \overline{KW} \) is the hypotenuse with a length of 6 units.
- \( \overline{KT} \) is one leg of the triangle with a length of 5 units.
- \( \overline{TW} \) forms the height of the triangle.
### Solution:
To find the measure of \( \angle K \), we use trigonometric functions. The angle \( \angle K \) can be calculated using the cosine function:
\[ \cos(\mathbf{\theta}) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
For \( \angle K \):
\[ \cos(K) = \frac{KT}{KW} = \frac{5}{6} \]
Using the inverse cosine function (arccos),
\[ K = \cos^{-1} \left( \frac{5}{6} \right) \]
### Steps to Calculate:
1. Calculate the arccosine value using a calculator.
2. Convert the result from degrees to the nearest minute.
### Calculation:
Using a scientific calculator,
\[ K \approx 33.5573^\circ \]
To convert to degrees and minutes:
\[ 33.5573^\circ = 33^\circ + 0.5573^\circ \]
Convert the decimal part to minutes:
\[ 0.5573^\circ \times 60 \approx 33.438 \text{ minutes} \]
Thus,
\[ K \approx 33^\circ 33' \]
### Final](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba437994-7eb2-41d2-bf90-86c7e632409d%2F5aa38755-f51e-4ff3-ae30-6b765d6e944a%2Fhgmfz9q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Problem: Right Triangle Angle Calculation**
**Problem Statement:**
In the diagram below of right triangle \(KTW\), \(KW = 6\), \(KT = 5\), and \(m\angle KTW = 90^\circ\).
\[
\begin{array}{ccccc}
&&W & \\
&&| & \\
&&| &6 \\
T& \underline{\blacksquare} &\hspace{0.5cm} & \_ \_ \_ \_ \_ \_ &_K\,5
\end{array}
\]
**Question:**
What is the measure of \( \angle K \), to the nearest minute?
### Explanation of Diagram:
The diagram represents a right-angled triangle \(KTW\) where:
- \(W\) is the vertex opposite the right angle.
- \( \overline{KW} \) is the hypotenuse with a length of 6 units.
- \( \overline{KT} \) is one leg of the triangle with a length of 5 units.
- \( \overline{TW} \) forms the height of the triangle.
### Solution:
To find the measure of \( \angle K \), we use trigonometric functions. The angle \( \angle K \) can be calculated using the cosine function:
\[ \cos(\mathbf{\theta}) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
For \( \angle K \):
\[ \cos(K) = \frac{KT}{KW} = \frac{5}{6} \]
Using the inverse cosine function (arccos),
\[ K = \cos^{-1} \left( \frac{5}{6} \right) \]
### Steps to Calculate:
1. Calculate the arccosine value using a calculator.
2. Convert the result from degrees to the nearest minute.
### Calculation:
Using a scientific calculator,
\[ K \approx 33.5573^\circ \]
To convert to degrees and minutes:
\[ 33.5573^\circ = 33^\circ + 0.5573^\circ \]
Convert the decimal part to minutes:
\[ 0.5573^\circ \times 60 \approx 33.438 \text{ minutes} \]
Thus,
\[ K \approx 33^\circ 33' \]
### Final
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