Solve the right triangle shown in the following figure.
80
A right triangle is given.
The first side of length 80 is opposite the angle 75°.
The second side is opposite the right angle.
The third side is opposite an unlabeled angle.
Find the length of the side adjacent to the given angle. (Round your answer to two decimal places.)
 
Find the length of the hypotenuse. (Round your answer to two decimal places.)
 
Find the other acute angle (in degrees).

To solve the right triangle, we can apply trigonometric functions and the Pythagorean theorem. Let's break down the problem step by step.

### 1. Find the length of the side adjacent to the given angle.
Given:
- The side opposite the angle \(75^\circ\) has a length of \(80\).
- We need to find the length of the side adjacent to the angle \(75^\circ\).

We can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
\[
\tan(75^\circ) = \frac{80}{\text{adjacent}}
\]
Let's solve for the adjacent side:
\[
\text{adjacent} = \frac{80}{\tan(75^\circ)}
\]

### 2. Find the length of the hypotenuse.
We can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
\[
\sin(75^\circ) = \frac{80}{\text{hypotenuse}}
\]
Let's solve for the hypotenuse:
\[
\text{hypotenuse} = \frac{80}{\sin(75^\circ)}
\]

### 3. Find the other acute angle.
In a right triangle, the sum of the angles is \(180^\circ\). Since one angle is \(90^\circ\) and another is \(75^\circ\), the third angle is:
\[
\text{Other acute angle} = 90^\circ - 75^\circ = 15^\circ
\]

Now, let's calculate these values.Here are the results:

1. **Length of the side adjacent to the given angle (75°):** \( 21.44 \) (rounded to two decimal places)
2. **Length of the hypotenuse:** \( 82.82 \) (rounded to two decimal places)
3. **The other acute angle:** \( 15^\circ \)

Would you like more details or have any questions? 

Here are some related questions to consider:
1. How do you derive the tangent and sine functions from a right triangle?
2. What other methods could be used to find the length of the hypotenuse?
3. How does the Pythagorean theorem relate to these calculations?
4. How would you solve the problem if a different angle or side were given?
5. What are some real-life applications of solving right triangles?

**Tip:** Always double-check your calculations, especially when working with angles and trigonometric functions. Small errors in input can lead to significant differences in results.

Solve the right triangle shown in the following figure. The first side of length 80 is opposite the angle 75°. The second side is opposite the right angle. The third side is opposite an unlabeled angle. Find the length of the side adjacent to the given angle. (Round your answer to two decimal places.) Find the length of the hypotenuse. (Round your answer to two decimal places.) Find the other acute angle (in degrees).

Learn how to solve a right triangle by finding the length of the side adjacent to a 75° angle, calculating the hypotenuse, and determining the other acute angle. This problem involves trigonometric functions and the Pythagorean theorem, suitable for students in Grades 9-12.

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