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The problem presents a right triangle with one angle of \(48^\circ\) and the length of one leg \(6\). The task is to find the values of \(\beta\), \(a\), and \(c\).

### Step 1: Find \(\beta\)
Since the triangle is a right triangle, we know that the sum of the angles must equal \(180^\circ\). Therefore, we can find \(\beta\) by subtracting the given angles from \(90^\circ\):

\[
\beta = 90^\circ - 48^\circ = 42^\circ
\]

### Step 2: Use Trigonometric Ratios
Now, we can use trigonometric ratios to find the lengths of sides \(a\) and \(c\).

- **Finding \(a\):**
Using the tangent function, which relates the opposite and adjacent sides of a right triangle:

\[
\tan(48^\circ) = \frac{6}{a}
\]

Solving for \(a\):

\[
a = \frac{6}{\tan(48^\circ)} \approx \frac{6}{1.1106} \approx 5.4
\]

- **Finding \(c\):**
Using the sine function, which relates the opposite side to the hypotenuse:

\[
\sin(48^\circ) = \frac{6}{c}
\]

Solving for \(c\):

\[
c = \frac{6}{\sin(48^\circ)} \approx \frac{6}{0.7431} \approx 8.1
\]

### Final Answers:
- \(\beta \approx 42^\circ\)
- \(a \approx 5.4\)
- \(c \approx 8.1\)

Would you like a deeper explanation for any of these steps?

Here are 5 related questions to expand this:
1. How can we use the Pythagorean theorem to verify the length of \(c\)?
2. What would the values of the sides be if the given leg was different?
3. How do trigonometric ratios change for different angles in a right triangle?
4. Can we use cosine instead of sine or tangent to solve this problem?
5. How would we solve for the unknowns if the given angle were \(\beta\) instead of \(48^\circ\)?

**Tip:** In right triangles, using trigonometric identities is a quick way to solve for unknown sides and angles without using more complicated methods.

Find β, a, and c for a right triangle with one angle of 48° and a known side length of 6 (adjacent to 48°).

In this problem, we solve for the missing angle β and the sides a and c of a right triangle. With one angle given as 48° and one side length known (6), we use trigonometric functions such as sine and tangent to find the unknown values. Suitable for students in Grades 9-10.

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