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 3: Finding the Values of \(u\) and \(t\) Using Sine and Cosine
Given a right triangle, find the lengths of sides \(u\) and \(t\). Round your answers to the nearest tenth.
Diagram Description:
- The triangle is a right-angled triangle.
- One angle is given as \( 65^\circ \).
- The hypotenuse is labeled as \(8\).
- The opposite side to the \(65^\circ\) angle is labeled as \(u\).
- The adjacent side to the \(65^\circ\) angle is labeled as \(t\).
### Steps to Solve:
1. **Using the Sine Function:**
\[
\sin(65^\circ) = \frac{u}{8}
\]
\[
u = 8 \cdot \sin(65^\circ)
\]
Using a calculator:
\[
u \approx 8 \cdot 0.9063 \approx 7.3
\]
2. **Using the Cosine Function:**
\[
\cos(65^\circ) = \frac{t}{8}
\]
\[
t = 8 \cdot \cos(65^\circ)
\]
Using a calculator:
\[
t \approx 8 \cdot 0.4226 \approx 3.4
\]
### Final Values:
- Length of \(u \approx 7.3\)
- Length of \(t \approx 3.4\)
Use trigonometric functions to find the values of the sides in a right-angled triangle with the given hypotenuse and angle. This can be a practical example to understand the application of sine and cosine in determining the lengths of triangle sides.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba437994-7eb2-41d2-bf90-86c7e632409d%2Fa7fbe7fd-0e9c-45b3-9137-1e70f7a6f427%2Fwspmpe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Finding the Values of \(u\) and \(t\) Using Sine and Cosine
Given a right triangle, find the lengths of sides \(u\) and \(t\). Round your answers to the nearest tenth.
Diagram Description:
- The triangle is a right-angled triangle.
- One angle is given as \( 65^\circ \).
- The hypotenuse is labeled as \(8\).
- The opposite side to the \(65^\circ\) angle is labeled as \(u\).
- The adjacent side to the \(65^\circ\) angle is labeled as \(t\).
### Steps to Solve:
1. **Using the Sine Function:**
\[
\sin(65^\circ) = \frac{u}{8}
\]
\[
u = 8 \cdot \sin(65^\circ)
\]
Using a calculator:
\[
u \approx 8 \cdot 0.9063 \approx 7.3
\]
2. **Using the Cosine Function:**
\[
\cos(65^\circ) = \frac{t}{8}
\]
\[
t = 8 \cdot \cos(65^\circ)
\]
Using a calculator:
\[
t \approx 8 \cdot 0.4226 \approx 3.4
\]
### Final Values:
- Length of \(u \approx 7.3\)
- Length of \(t \approx 3.4\)
Use trigonometric functions to find the values of the sides in a right-angled triangle with the given hypotenuse and angle. This can be a practical example to understand the application of sine and cosine in determining the lengths of triangle sides.
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