https://file.aiquickdraw.com/mathbot/file/1725045412710ftbar33b.jpg


Solve the side length a
, as we know that b=7.20 cm
 and the angles marked in the figure are α=25.68∘
 and β=33.06∘
.

To solve for the side length \( a \) in the given diagram, we can use trigonometric relationships. Here's the step-by-step process:

### Step 1: Identify the relevant angles and sides.
From the diagram:
- \( \alpha = 25.68^\circ \)
- \( \beta = 33.06^\circ \)
- \( b = 7.20 \) cm

### Step 2: Use the tangent function.
We can use the tangent function in relation to the angles \( \alpha \) and \( \beta \):

For \( \alpha \):
\[
\tan(\alpha) = \frac{b}{a}
\]
\[
\tan(25.68^\circ) = \frac{7.20 \text{ cm}}{a}
\]

### Step 3: Solve for \( a \).
First, calculate the value of \( \tan(25.68^\circ) \):
\[
\tan(25.68^\circ) \approx 0.4803
\]

Now, use the tangent function to solve for \( a \):
\[
a = \frac{7.20 \text{ cm}}{0.4803} \approx 14.99 \text{ cm}
\]

So, the side length \( a \) is approximately \( 14.99 \) cm.

Would you like more details or have any questions?

### Related Questions:
1. How would you calculate side \( b \) if \( a \) and the angles were known?
2. How can you determine the length of the hypotenuse of the right triangle formed by \( a \) and \( b \)?
3. What other trigonometric functions could be used to find \( a \) or \( b \)?
4. How does changing \( \alpha \) affect the side length \( a \)?
5. How would the solution change if angle \( \alpha \) or \( \beta \) was greater than \( 45^\circ \)?

### Tip:
Always double-check the angle units (degrees vs. radians) in your calculations to ensure accuracy.

Learn how to solve for the side length a in a right triangle using trigonometric functions. Given angles α = 25.68°, β = 33.06°, and side length b = 7.20 cm, find the value of a. Detailed step-by-step solution included.

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