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:**
In triangle ΔQRS, the measure of ∠S = 90°, \( QR = 71 \) feet, and \( SQ = 50 \) feet. Find the measure of ∠R to the nearest degree.
**Diagram Explanation:**
The given diagram is of a right triangle labeled ΔQRS with:
- A right angle at point S.
- The hypotenuse QR measured at 71 feet.
- One leg SQ measured at 50 feet.
**Steps to Solve:**
1. To find ∠R, we can use trigonometric ratios. Considering the right triangle properties:
- The side opposite to ∠R is SQ.
- The hypotenuse is QR.
2. We use the sine trigonometric ratio:
\[
\sin(x^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{SQ}{QR}
\]
\[
\sin(R) = \frac{50}{71}
\]
3. Calculate the sine value:
\[
\sin(R) = \frac{50}{71} \approx 0.7042
\]
4. To find ∠R, take the inverse sine (arcsine) of 0.7042:
\[
R = \sin^{-1}(0.7042)
\]
5. Using a calculator:
\[
R \approx 44.98^\circ
\]
6. To the nearest degree:
\[
R \approx 45^\circ
\]
**Answer:**
The measure of ∠R in ΔQRS, to the nearest degree, is 45°.
**Trigonometric Insight:**
This problem uses sine, one of the fundamental trigonometric functions which relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. This is crucial in solving many geometric and real-world measurement problems involving right triangles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1ee66a4-8e77-4cb9-b1b9-9bd40d747386%2Fe81f164a-c0d9-45af-b847-8a683f334c48%2Fu298ccq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
In triangle ΔQRS, the measure of ∠S = 90°, \( QR = 71 \) feet, and \( SQ = 50 \) feet. Find the measure of ∠R to the nearest degree.
**Diagram Explanation:**
The given diagram is of a right triangle labeled ΔQRS with:
- A right angle at point S.
- The hypotenuse QR measured at 71 feet.
- One leg SQ measured at 50 feet.
**Steps to Solve:**
1. To find ∠R, we can use trigonometric ratios. Considering the right triangle properties:
- The side opposite to ∠R is SQ.
- The hypotenuse is QR.
2. We use the sine trigonometric ratio:
\[
\sin(x^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{SQ}{QR}
\]
\[
\sin(R) = \frac{50}{71}
\]
3. Calculate the sine value:
\[
\sin(R) = \frac{50}{71} \approx 0.7042
\]
4. To find ∠R, take the inverse sine (arcsine) of 0.7042:
\[
R = \sin^{-1}(0.7042)
\]
5. Using a calculator:
\[
R \approx 44.98^\circ
\]
6. To the nearest degree:
\[
R \approx 45^\circ
\]
**Answer:**
The measure of ∠R in ΔQRS, to the nearest degree, is 45°.
**Trigonometric Insight:**
This problem uses sine, one of the fundamental trigonometric functions which relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. This is crucial in solving many geometric and real-world measurement problems involving right triangles.
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