Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter2: Right Triangle Trigonometry
Section2.4: Applications
Problem 48PS: Albert lives in New Orleans. At noon on a summer day, the angle of elevation of the sun is 84. The...
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![Title: Finding the Sine of an Angle in a Right Triangle
**Objective:**
Learn how to find the sine of an angle in a right triangle and simplify your answer.
**Problem:**
Find the sine of ∠Y in the given right triangle XYZ where:
- The side opposite ∠Y (YZ) measures 10 units.
- The side adjacent to ∠Y (ZX) measures 24 units.
- The hypotenuse (XY) needs to be calculated.
**Visual Representation:**
The triangle is illustrated with vertices labeled Y, Z, and X. The angle at Y is the angle of interest (∠Y). The right angle is at Z.
**Triangle Sides and Angle:**
- \( YZ = 10 \) units (opposite to ∠Y)
- \( XZ = 24 \) units (adjacent to ∠Y)
- \( XY \) (hypotenuse): To be calculated using the Pythagorean theorem.
**Steps:**
1. **Calculate the hypotenuse (XY):**
Use the Pythagorean theorem: \( XY = \sqrt{(YZ)^2 + (XZ)^2} \)
\[
XY = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26
\]
2. **Find the sine of ∠Y:**
The sine of an angle in a right triangle is given by the ratio of the length of the opposite side to the hypotenuse.
\[
\sin(Y) = \frac{YZ}{XY} = \frac{10}{26} = \frac{5}{13}
\]
3. **Simplify and present your answer:**
The simplified form of \( \sin(Y) = \frac{5}{13} \).
**Answer:**
\[
\sin(Y) = \frac{5}{13}
\]
Ensure to provide your answer as a proper fraction, improper fraction, or whole number as required.
**Interactive Component:**
An input box is provided to enter the final answer, with options for fractional and radical formats to confirm accuracy.
**Conclusion:**
Understanding how to find sine values in right triangles is essential for trigonometry and can simplify solving geometric problems efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa89e63f8-66db-4f4d-a2d5-5c1b9146f99d%2Fd1c5f892-b24c-462e-b6e5-308177ce4fa2%2Frlg2rku_processed.png&w=3840&q=75)
Transcribed Image Text:Title: Finding the Sine of an Angle in a Right Triangle
**Objective:**
Learn how to find the sine of an angle in a right triangle and simplify your answer.
**Problem:**
Find the sine of ∠Y in the given right triangle XYZ where:
- The side opposite ∠Y (YZ) measures 10 units.
- The side adjacent to ∠Y (ZX) measures 24 units.
- The hypotenuse (XY) needs to be calculated.
**Visual Representation:**
The triangle is illustrated with vertices labeled Y, Z, and X. The angle at Y is the angle of interest (∠Y). The right angle is at Z.
**Triangle Sides and Angle:**
- \( YZ = 10 \) units (opposite to ∠Y)
- \( XZ = 24 \) units (adjacent to ∠Y)
- \( XY \) (hypotenuse): To be calculated using the Pythagorean theorem.
**Steps:**
1. **Calculate the hypotenuse (XY):**
Use the Pythagorean theorem: \( XY = \sqrt{(YZ)^2 + (XZ)^2} \)
\[
XY = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26
\]
2. **Find the sine of ∠Y:**
The sine of an angle in a right triangle is given by the ratio of the length of the opposite side to the hypotenuse.
\[
\sin(Y) = \frac{YZ}{XY} = \frac{10}{26} = \frac{5}{13}
\]
3. **Simplify and present your answer:**
The simplified form of \( \sin(Y) = \frac{5}{13} \).
**Answer:**
\[
\sin(Y) = \frac{5}{13}
\]
Ensure to provide your answer as a proper fraction, improper fraction, or whole number as required.
**Interactive Component:**
An input box is provided to enter the final answer, with options for fractional and radical formats to confirm accuracy.
**Conclusion:**
Understanding how to find sine values in right triangles is essential for trigonometry and can simplify solving geometric problems efficiently.
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