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
Which trig ratio would you use to solve for the missing angle measures
![**Trigonometry Problem - Solving for Angles**
In this question, we will explore how to solve for a missing angle in a right triangle using trigonometric ratios.
**Question 7/10**
**Part A**
Given a right triangle:
- The side opposite the angle \( x \) has a length of 3 units.
- The hypotenuse has a length of 25 units.
Which trigonometric ratio would you use to solve for the missing angle measure?
Options:
A) \( \cos^{-1} \left( \frac{3}{25} \right) \)
B) \( \cos^{-1} \left( \frac{25}{3} \right) \)
C) \( \sin^{-1} \left( \frac{3}{25} \right) \)
D) \( \sin^{-1} \left( \frac{25}{3} \right) \)
**Part B**
Find the measure of angle \( x \). Round the answer to the nearest whole number.
---
**Explanation for Diagrams and Graphs:**
The diagram provided illustrates a right triangle with one angle labeled as \( x^\circ \):
- One leg of the triangle (adjacent to the right angle) measures 3 units.
- The hypotenuse of the triangle measures 25 units.
To solve for the angle \( x \), you would use the sine trigonometric function, \( \sin(x) \), because it involves the ratio of the length of the opposite side to the hypotenuse:
\[ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{25} \]
Thus, you would use the inverse sine function to find angle \( x \):
\[ x = \sin^{-1} \left( \frac{3}{25} \right) \]
After solving for \( x \) using a calculator:
\[ x \approx \sin^{-1} \left( 0.12 \right) \approx 7^\circ \]
Therefore, the correct answers are:
**Part A:** Answer C) \( \sin^{-1} \left( \frac{3}{25} \right) \)
**Part B:** The measure of angle \( x \) is approximately 7 degrees.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F132ab8cc-7ff2-4fb9-ae8b-a604bfc1e184%2F8853d945-0c2f-40d8-bd79-49bb0904d8b2%2Fxypk3t5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometry Problem - Solving for Angles**
In this question, we will explore how to solve for a missing angle in a right triangle using trigonometric ratios.
**Question 7/10**
**Part A**
Given a right triangle:
- The side opposite the angle \( x \) has a length of 3 units.
- The hypotenuse has a length of 25 units.
Which trigonometric ratio would you use to solve for the missing angle measure?
Options:
A) \( \cos^{-1} \left( \frac{3}{25} \right) \)
B) \( \cos^{-1} \left( \frac{25}{3} \right) \)
C) \( \sin^{-1} \left( \frac{3}{25} \right) \)
D) \( \sin^{-1} \left( \frac{25}{3} \right) \)
**Part B**
Find the measure of angle \( x \). Round the answer to the nearest whole number.
---
**Explanation for Diagrams and Graphs:**
The diagram provided illustrates a right triangle with one angle labeled as \( x^\circ \):
- One leg of the triangle (adjacent to the right angle) measures 3 units.
- The hypotenuse of the triangle measures 25 units.
To solve for the angle \( x \), you would use the sine trigonometric function, \( \sin(x) \), because it involves the ratio of the length of the opposite side to the hypotenuse:
\[ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{25} \]
Thus, you would use the inverse sine function to find angle \( x \):
\[ x = \sin^{-1} \left( \frac{3}{25} \right) \]
After solving for \( x \) using a calculator:
\[ x \approx \sin^{-1} \left( 0.12 \right) \approx 7^\circ \]
Therefore, the correct answers are:
**Part A:** Answer C) \( \sin^{-1} \left( \frac{3}{25} \right) \)
**Part B:** The measure of angle \( x \) is approximately 7 degrees.
Transcribed Image Text:**Question 7/10**
**Diagram:**
- A right-angled triangle is shown. The side opposite the right angle (hypotenuse) is labeled as 25.
- The side adjacent to angle \( x^\circ \) is labeled as 3.
- Another side is unlabeled.
### Part A
**Which trig ratio would you use to solve for the missing angle measure?**
A) \(\cos^{-1}\left(\dfrac{3}{25}\right)\)
B) \(\cos^{-1}\left(\dfrac{25}{3}\right)\)
C) \(\sin^{-1}\left(\dfrac{3}{25}\right)\)
D) \(\sin^{-1}\left(\dfrac{25}{3}\right)\)
### Part B
**Find the measure of angle \( x \). Round answer to nearest whole number.**
(Answer box provided)
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