Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
ChapterCSR: Contents Of Student Resources
Section: Chapter Questions
Problem 1.39EP
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Question
![### Geometry Problem: Finding the Length of a Side
The image features a right triangle \( \triangle STU \). Below are the details and the diagram of the triangle:
**Diagram Description:**
- The triangle \( \triangle STU \) is depicted with:
- Angle \( \angle STU = 21^\circ \)
- \( ST = x \)
- \( TU = 5.5 \)
- \( \angle T = 90^\circ \)
**Purpose:**
To solve problems involving the sides and angles of right triangles, specifically focusing on trigonometric relationships.
**Explanation:**
Given:
1. A right triangle \( \triangle STU \)
2. \( \angle STU = 21^\circ \)
3. \( \text{Length of } TU = 5.5 \)
4. \( \text{Length of } ST = x \)
5. \( \angle T = 90^\circ \)
### Trigonometric Relationship:
To find the unknown length \( x \), we can use the trigonometric ratios, specifically the tangent (tan) function since we know an angle and the length of the side adjacent to it.
\[ \tan(21^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(21^\circ) = \frac{ST}{TU} \]
\[ \tan(21^\circ) = \frac{x}{5.5} \]
From the equation, solve for \( x \):
\[ x = 5.5 \times \tan(21^\circ) \]
Using a calculator, find \( \tan(21^\circ) \) and then multiply by 5.5 to find the length of \( ST \).
This process demonstrates how to utilize the tangent function to determine unknown side lengths in right triangles using known angles and side lengths.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f6f557d-9e1e-46fd-a63e-879d74571691%2Fa419d321-bb5c-4816-98a0-7b18968acb2a%2Fh9cq6at_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Finding the Length of a Side
The image features a right triangle \( \triangle STU \). Below are the details and the diagram of the triangle:
**Diagram Description:**
- The triangle \( \triangle STU \) is depicted with:
- Angle \( \angle STU = 21^\circ \)
- \( ST = x \)
- \( TU = 5.5 \)
- \( \angle T = 90^\circ \)
**Purpose:**
To solve problems involving the sides and angles of right triangles, specifically focusing on trigonometric relationships.
**Explanation:**
Given:
1. A right triangle \( \triangle STU \)
2. \( \angle STU = 21^\circ \)
3. \( \text{Length of } TU = 5.5 \)
4. \( \text{Length of } ST = x \)
5. \( \angle T = 90^\circ \)
### Trigonometric Relationship:
To find the unknown length \( x \), we can use the trigonometric ratios, specifically the tangent (tan) function since we know an angle and the length of the side adjacent to it.
\[ \tan(21^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(21^\circ) = \frac{ST}{TU} \]
\[ \tan(21^\circ) = \frac{x}{5.5} \]
From the equation, solve for \( x \):
\[ x = 5.5 \times \tan(21^\circ) \]
Using a calculator, find \( \tan(21^\circ) \) and then multiply by 5.5 to find the length of \( ST \).
This process demonstrates how to utilize the tangent function to determine unknown side lengths in right triangles using known angles and side lengths.
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