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...
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![### Solving for \( x \) in a Right Triangle
In this exercise, you are given a right triangle and asked to solve for the angle \( x \).
#### Instructions:
Solve for \( x \). Round to the nearest tenth of a degree, if necessary.
#### Given Diagram:
The triangle is labeled as follows:
- \( \angle I \) is the right angle.
- Side \( IH \) is the hypotenuse and measures 59 units.
- Side \( GI \) measures 50 units.
- \( \angle GIH \) is given as \( x^\circ \).
\( x \) is the unknown angle we need to solve for.
#### Solution Explanation:
To find \( x \), we can use trigonometric ratios. Here, we have the length of the side adjacent to \( x \) (which is 50) and the hypotenuse (which is 59).
We can use the cosine function, which is defined as:
\[ \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substitute the given values:
\[ \cos(x) = \frac{50}{59} \]
To find \( x \), take the arccosine (inverse cosine) of both sides:
\[ x = \cos^{-1}\left(\frac{50}{59}\right) \]
Use a calculator to find the value:
\[ x \approx 32.2^\circ \]
#### Final Answer:
\[ x = 32.2^\circ \]
Enter your answer in the provided answer box labeled "Answer: \( x = \)". Then, click "Submit Answer" to check your solution.
#### Additional Notes:
You may use a scientific calculator to compute the arccosine value. If necessary, refer to materials on trigonometric functions and their inverses for deeper understanding.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13a39696-0c18-4420-97eb-04a57a421d8a%2F6dbfd1a0-8bf5-41c2-a70d-800c4a20c6a4%2Fuakqxad_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving for \( x \) in a Right Triangle
In this exercise, you are given a right triangle and asked to solve for the angle \( x \).
#### Instructions:
Solve for \( x \). Round to the nearest tenth of a degree, if necessary.
#### Given Diagram:
The triangle is labeled as follows:
- \( \angle I \) is the right angle.
- Side \( IH \) is the hypotenuse and measures 59 units.
- Side \( GI \) measures 50 units.
- \( \angle GIH \) is given as \( x^\circ \).
\( x \) is the unknown angle we need to solve for.
#### Solution Explanation:
To find \( x \), we can use trigonometric ratios. Here, we have the length of the side adjacent to \( x \) (which is 50) and the hypotenuse (which is 59).
We can use the cosine function, which is defined as:
\[ \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substitute the given values:
\[ \cos(x) = \frac{50}{59} \]
To find \( x \), take the arccosine (inverse cosine) of both sides:
\[ x = \cos^{-1}\left(\frac{50}{59}\right) \]
Use a calculator to find the value:
\[ x \approx 32.2^\circ \]
#### Final Answer:
\[ x = 32.2^\circ \]
Enter your answer in the provided answer box labeled "Answer: \( x = \)". Then, click "Submit Answer" to check your solution.
#### Additional Notes:
You may use a scientific calculator to compute the arccosine value. If necessary, refer to materials on trigonometric functions and their inverses for deeper understanding.
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