K L 6.5 4.4 to M
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.2: The Law Of Cosines
Problem 3E
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Transcribed Image Text:This image contains a right-angled triangle KLM. The vertices of the triangle are labeled K, L, and M, with angle L being the right angle.
- Segment LM is the base of the triangle and measures 4.4 units.
- Segment LK is the height of the triangle, but the length is not specified.
- Segment KM is the hypotenuse of the triangle and measures 6.5 units.
- There is an angle marked as \( x^\circ \) at vertex M.
To find the values related to the triangle, such as the unknown side length or the angle \( x^\circ \), one might use the Pythagorean theorem or trigonometric ratios.
![**Understanding Right Triangles: Example Problem**
In this educational segment, we present an illustrated example of a right-angled triangle to aid in your understanding of basic trigonometric principles and properties.
### Diagram Description:
The diagram displays a right-angled triangle \(\triangle KLM\):
- **Vertices**: The vertices of the triangle are labeled as \(K\), \(L\), and \(M\).
- **Right Angle**: Vertex \(L\) is the right angle (\(90^\circ\)).
- **Sides**:
- \(LM\) is the base of the triangle with a length of \(4.4\) units.
- \(KL\) is the perpendicular side.
- \(KM\) is the hypotenuse, the side opposite the right angle, with a length of \(6.5\) units.
- **Angle**: The angle at vertex \(M\) is denoted as \(x^\circ\).
### Concept Explanation:
In a right triangle, such as \(\triangle KLM\) in the diagram, the relationship between the sides and angles can be understood using trigonometric ratios and the Pythagorean theorem.
**Key Points to Remember:**
1. **Pythagorean Theorem**: For a right triangle, the square of the hypotenuse (\(KM\)) is equal to the sum of the squares of the other two sides (\(KL\) and \(LM\)).
\[
(KL)^2 + (LM)^2 = (KM)^2
\]
2. **Trigonometric Ratios**:
- **Sine of angle \(x\)**: \(\sin(x) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{KL}{KM}\)
- **Cosine of angle \(x\)**: \(\cos(x) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{LM}{KM}\)
- **Tangent of angle \(x\)**: \(\tan(x) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{KL}{LM}\)
Using the given measures and these principles, various properties and unknown values of the triangle can be calculated.
### Problem-Solving Examples:
Using the provided diagram, you can solve for unknown measurements](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83aba2ed-94d5-4425-b19d-68ed9b675f65%2F99a0b1b2-c475-46c8-b06c-dc44f3c33068%2F43mj0r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Right Triangles: Example Problem**
In this educational segment, we present an illustrated example of a right-angled triangle to aid in your understanding of basic trigonometric principles and properties.
### Diagram Description:
The diagram displays a right-angled triangle \(\triangle KLM\):
- **Vertices**: The vertices of the triangle are labeled as \(K\), \(L\), and \(M\).
- **Right Angle**: Vertex \(L\) is the right angle (\(90^\circ\)).
- **Sides**:
- \(LM\) is the base of the triangle with a length of \(4.4\) units.
- \(KL\) is the perpendicular side.
- \(KM\) is the hypotenuse, the side opposite the right angle, with a length of \(6.5\) units.
- **Angle**: The angle at vertex \(M\) is denoted as \(x^\circ\).
### Concept Explanation:
In a right triangle, such as \(\triangle KLM\) in the diagram, the relationship between the sides and angles can be understood using trigonometric ratios and the Pythagorean theorem.
**Key Points to Remember:**
1. **Pythagorean Theorem**: For a right triangle, the square of the hypotenuse (\(KM\)) is equal to the sum of the squares of the other two sides (\(KL\) and \(LM\)).
\[
(KL)^2 + (LM)^2 = (KM)^2
\]
2. **Trigonometric Ratios**:
- **Sine of angle \(x\)**: \(\sin(x) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} = \frac{KL}{KM}\)
- **Cosine of angle \(x\)**: \(\cos(x) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{LM}{KM}\)
- **Tangent of angle \(x\)**: \(\tan(x) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} = \frac{KL}{LM}\)
Using the given measures and these principles, various properties and unknown values of the triangle can be calculated.
### Problem-Solving Examples:
Using the provided diagram, you can solve for unknown measurements
![**Title: Solving for Angles in Right Triangles Using Trigonometry**
---
**Objective:**
Learn how to determine the angle of a right triangle when given two sides.
---
**Problem Statement:**
Given a right triangle \( KLM \) with:
- \( KL \) as the side opposite to angle \( x^\circ \)
- \( KM \) as the hypotenuse
- \( LM \) as the adjacent side to angle \( x^\circ \)
We are to solve for \( x \) and round to the nearest tenth of a degree, if necessary.
**Details of the Triangle:**
\[ \text{Side \( KM \), hypotenuse} = 6.5 \]
\[ \text{Side \( LM \), adjacent} = 4.4 \]
---
**Diagram Explanation:**
The triangle \( KLM \) is a right triangle. It includes a right angle at \( L \). Side \( KL \) is the side opposite to the angle \( x^\circ \), side \( LM \) is the adjacent side, and side \( KM \) is the hypotenuse of the triangle.
**Diagram:**
```
K
|\
| \
| \
| \
| \
6.5 x°
| \
|_______\
L 4.4 M
```
---
**Steps to Solve for \( x \):**
1. **Identify the Trigonometric Ratio to Use:**
Since the lengths of the adjacent side (LM) and the hypotenuse (KM) are given, we will use the cosine function.
\[ \cos(x^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{LM}{KM} \]
2. **Substitute the Given Values:**
\[ \cos(x^\circ) = \frac{4.4}{6.5} \]
3. **Calculate the Value:**
\[ \cos(x^\circ) = 0.6769 \]
4. **Find the Angle \( x \) Using an Inverse Cosine Function:**
\[ x = \cos^{-1}(0.6769) \]
5. **Compute the Result:**
Use a scientific calculator to find \( x \):
\[ x \approx](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83aba2ed-94d5-4425-b19d-68ed9b675f65%2F99a0b1b2-c475-46c8-b06c-dc44f3c33068%2Fsekox2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving for Angles in Right Triangles Using Trigonometry**
---
**Objective:**
Learn how to determine the angle of a right triangle when given two sides.
---
**Problem Statement:**
Given a right triangle \( KLM \) with:
- \( KL \) as the side opposite to angle \( x^\circ \)
- \( KM \) as the hypotenuse
- \( LM \) as the adjacent side to angle \( x^\circ \)
We are to solve for \( x \) and round to the nearest tenth of a degree, if necessary.
**Details of the Triangle:**
\[ \text{Side \( KM \), hypotenuse} = 6.5 \]
\[ \text{Side \( LM \), adjacent} = 4.4 \]
---
**Diagram Explanation:**
The triangle \( KLM \) is a right triangle. It includes a right angle at \( L \). Side \( KL \) is the side opposite to the angle \( x^\circ \), side \( LM \) is the adjacent side, and side \( KM \) is the hypotenuse of the triangle.
**Diagram:**
```
K
|\
| \
| \
| \
| \
6.5 x°
| \
|_______\
L 4.4 M
```
---
**Steps to Solve for \( x \):**
1. **Identify the Trigonometric Ratio to Use:**
Since the lengths of the adjacent side (LM) and the hypotenuse (KM) are given, we will use the cosine function.
\[ \cos(x^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{LM}{KM} \]
2. **Substitute the Given Values:**
\[ \cos(x^\circ) = \frac{4.4}{6.5} \]
3. **Calculate the Value:**
\[ \cos(x^\circ) = 0.6769 \]
4. **Find the Angle \( x \) Using an Inverse Cosine Function:**
\[ x = \cos^{-1}(0.6769) \]
5. **Compute the Result:**
Use a scientific calculator to find \( x \):
\[ x \approx
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