Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter7: Triangles
Section7.3: The Ambiguous Case
Problem 30PS: True Course A plane headed due east is traveling with an airspeed of 190 miles per hour. The wind...
Related questions
Question
In this problem, we are given a right triangle with the following details:
- One angle measures 30°.
- The hypotenuse of the triangle is 40 units long.
- The sides opposite and adjacent to the 30° angle are labeled with variables \(x\) and \(y\), respectively.
We are required to find the values of \(x\) and \(y\).
The diagram provided shows:
- The hypotenuse is labeled as 40.
- The angle at the base is labeled as 30°.
- The side opposite the 30° angle is labeled as \(x\).
- The side adjacent to the 30° angle is labeled as \(y\).
### Solutions
Using the properties of a 30°-60°-90° triangle:
- The side opposite the 30° angle (shorter leg) is half the length of the hypotenuse.
- The side opposite the 60° angle (longer leg) is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.
Given:
- Hypotenuse = 40 units
Thus:
\[ x = \frac{1}{2} \times 40 = 20 \]
\[ y = \frac{\sqrt{3}}{2} \times 40 = 20\sqrt{3} \]
### Final Values
So, the values are:
\[ x = 20 \]
\[ y = 20\sqrt{3} \]
(Simplify your answers. Type exact answers, using radicals as needed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F959a643c-45ff-4cba-bf81-f285af0e5e62%2F6e98547d-4ee4-4d58-b012-cd33ccb452e3%2Fdn2ils8_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the value of each variable.
In this problem, we are given a right triangle with the following details:
- One angle measures 30°.
- The hypotenuse of the triangle is 40 units long.
- The sides opposite and adjacent to the 30° angle are labeled with variables \(x\) and \(y\), respectively.
We are required to find the values of \(x\) and \(y\).
The diagram provided shows:
- The hypotenuse is labeled as 40.
- The angle at the base is labeled as 30°.
- The side opposite the 30° angle is labeled as \(x\).
- The side adjacent to the 30° angle is labeled as \(y\).
### Solutions
Using the properties of a 30°-60°-90° triangle:
- The side opposite the 30° angle (shorter leg) is half the length of the hypotenuse.
- The side opposite the 60° angle (longer leg) is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.
Given:
- Hypotenuse = 40 units
Thus:
\[ x = \frac{1}{2} \times 40 = 20 \]
\[ y = \frac{\sqrt{3}}{2} \times 40 = 20\sqrt{3} \]
### Final Values
So, the values are:
\[ x = 20 \]
\[ y = 20\sqrt{3} \]
(Simplify your answers. Type exact answers, using radicals as needed.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 1 images
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Elementary Algebra
Algebra
ISBN:
9780998625713
Author:
Lynn Marecek, MaryAnne Anthony-Smith
Publisher:
OpenStax - Rice University
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage