Find the exact values of all six trigonometric functions for the angle 0 of the right triangle. The right triangle has a hypotenuse of 15 and a leg side-length of 6 with the angle 0 in between the two given side lengths. Be sure to reduce all fractions, simplify all radicals, and have no decimals. 15
Find the exact values of all six trigonometric functions for the angle 0 of the right triangle. The right triangle has a hypotenuse of 15 and a leg side-length of 6 with the angle 0 in between the two given side lengths. Be sure to reduce all fractions, simplify all radicals, and have no decimals. 15
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
![**Trigonometric Functions for a Right Triangle**
Find the exact values of all six trigonometric functions for the angle \( \theta \) of the right triangle. The right triangle has a hypotenuse of 15 and a leg side-length of 6 with the angle \( \theta \) in between the two given side lengths. **Be sure to reduce all fractions, simplify all radicals, and have no decimals.**
**Diagram Explanation:**
The diagram shows a right triangle with:
- A hypotenuse labeled as 15.
- A horizontal leg labeled as 6.
- The angle \( \theta \) between the hypotenuse and the horizontal leg.
- The vertical leg is not labeled, but is perpendicular to the horizontal leg.
**Steps to Solve:**
1. **Calculate the missing side:**
Use the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Let \( a = 6 \), \( b \) be the missing side, and \( c = 15 \).
\[
6^2 + b^2 = 15^2 \\
36 + b^2 = 225 \\
b^2 = 189 \\
b = \sqrt{189} = 3\sqrt{21}
\]
2. **Find the trigonometric functions:**
- **Sine** (\( \sin \theta \)): \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3\sqrt{21}}{15} = \frac{\sqrt{21}}{5}\)
- **Cosine** (\( \cos \theta \)): \(\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{15} = \frac{2}{5}\)
- **Tangent** (\( \tan \theta \)): \(\frac{\text{opposite}}{\text{adjacent}} = \frac{3\sqrt{21}}{6} = \frac{\sqrt{21}}{2}\)
- **Cosecant** (\( \csc \theta \)): \(\frac{\text{hypotenuse}}{\text{opposite}} = \frac{15}{3\sqrt{21}} = \frac{5}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33693f5b-e996-45d6-9361-e39b55dcab94%2Fc6757502-b446-4526-9e6d-04d31d3eeffd%2F2eaq83_processed.png&w=3840&q=75)
Transcribed Image Text:**Trigonometric Functions for a Right Triangle**
Find the exact values of all six trigonometric functions for the angle \( \theta \) of the right triangle. The right triangle has a hypotenuse of 15 and a leg side-length of 6 with the angle \( \theta \) in between the two given side lengths. **Be sure to reduce all fractions, simplify all radicals, and have no decimals.**
**Diagram Explanation:**
The diagram shows a right triangle with:
- A hypotenuse labeled as 15.
- A horizontal leg labeled as 6.
- The angle \( \theta \) between the hypotenuse and the horizontal leg.
- The vertical leg is not labeled, but is perpendicular to the horizontal leg.
**Steps to Solve:**
1. **Calculate the missing side:**
Use the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Let \( a = 6 \), \( b \) be the missing side, and \( c = 15 \).
\[
6^2 + b^2 = 15^2 \\
36 + b^2 = 225 \\
b^2 = 189 \\
b = \sqrt{189} = 3\sqrt{21}
\]
2. **Find the trigonometric functions:**
- **Sine** (\( \sin \theta \)): \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3\sqrt{21}}{15} = \frac{\sqrt{21}}{5}\)
- **Cosine** (\( \cos \theta \)): \(\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{15} = \frac{2}{5}\)
- **Tangent** (\( \tan \theta \)): \(\frac{\text{opposite}}{\text{adjacent}} = \frac{3\sqrt{21}}{6} = \frac{\sqrt{21}}{2}\)
- **Cosecant** (\( \csc \theta \)): \(\frac{\text{hypotenuse}}{\text{opposite}} = \frac{15}{3\sqrt{21}} = \frac{5}{
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