Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Concept explainers
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
Find the value of each trigonometric ratio to the nearest ten-thousandth.
![**Trigonometry Problem: Calculating Cosine**
**Problem Statement:**
Given the right triangle \( \triangle XYZ \) with side lengths \( XY = 12 \) units, \( YZ = 35 \) units, and hypotenuse \( XZ = 37 \) units, find \(\cos Z\).
**Diagram:**
The diagram presents a right triangle \( \triangle XYZ \) where:
- \( X \) and \( Y \) are the two vertices forming the right angle.
- Side \( XY \) (adjacent to angle \( Z \)) has a length of 12 units.
- Side \( YZ \) (opposite to angle \( Z \)) has a length of 35 units.
- Hypotenuse \( XZ \) has a length of 37 units.
**Multiple Choice Options:**
A) 0.3429
B) 0.9459
C) 0.3243
D) 0.2167 (crossed out)
**Explanation:**
To find \(\cos Z\), we utilize the definition of cosine in a right triangle, which is given by the ratio of the length of the adjacent side to the length of the hypotenuse.
\[
\cos Z = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{XY}{XZ} = \frac{12}{37}
\]
Calculating the ratio:
\[
\frac{12}{37} \approx 0.3243
\]
Thus, the correct answer is:
**C) 0.3243**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaacf69e-0331-4d40-b25d-0435759925c0%2F088223aa-d5fb-446d-b094-cf6909dbad0b%2F1302gi_processed.png&w=3840&q=75)
Transcribed Image Text:**Trigonometry Problem: Calculating Cosine**
**Problem Statement:**
Given the right triangle \( \triangle XYZ \) with side lengths \( XY = 12 \) units, \( YZ = 35 \) units, and hypotenuse \( XZ = 37 \) units, find \(\cos Z\).
**Diagram:**
The diagram presents a right triangle \( \triangle XYZ \) where:
- \( X \) and \( Y \) are the two vertices forming the right angle.
- Side \( XY \) (adjacent to angle \( Z \)) has a length of 12 units.
- Side \( YZ \) (opposite to angle \( Z \)) has a length of 35 units.
- Hypotenuse \( XZ \) has a length of 37 units.
**Multiple Choice Options:**
A) 0.3429
B) 0.9459
C) 0.3243
D) 0.2167 (crossed out)
**Explanation:**
To find \(\cos Z\), we utilize the definition of cosine in a right triangle, which is given by the ratio of the length of the adjacent side to the length of the hypotenuse.
\[
\cos Z = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{XY}{XZ} = \frac{12}{37}
\]
Calculating the ratio:
\[
\frac{12}{37} \approx 0.3243
\]
Thus, the correct answer is:
**C) 0.3243**
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