Understanding the Problem
We are given a right-angled triangle. In such a triangle, one angle is always \(90^\circ\). The other two angles must be acute (less than \(90^\circ\)). We are told that the difference between these two acute angles is \(\frac{\pi}{12}\) radians. Our goal is to find the measure of one of these acute angles.
Key Properties of a Right-Angled Triangle
- The sum of all angles in any triangle is \(180^\circ\).
- In a right-angled triangle, one angle is \(90^\circ\).
- Therefore, the sum of the two acute angles in a right-angled triangle is \(180^\circ - 90^\circ = 90^\circ\).
Calculations
Let the two acute angles be \(\alpha\) and \(\beta\). From the properties above, we have:
\(\alpha + \beta = 90^\circ\)
We are given the difference between these angles is \(\frac{\pi}{12}\) radians. First, let's convert this difference to degrees:
\(\frac{\pi}{12} \text{ radians} = \frac{\pi}{12} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{12} = 15^\circ\)
So, the difference between the two acute angles is \(15^\circ\). We can write this as:
\(|\alpha - \beta| = 15^\circ\)
Let's assume \(\alpha\) is the larger acute angle. Then:
\(\alpha - \beta = 15^\circ\)
Now we have a system of two linear equations:
- \(\alpha + \beta = 90^\circ\)
- \(\alpha - \beta = 15^\circ\)
To find the value of \(\alpha\), we can add the two equations:
\((\alpha + \beta) + (\alpha - \beta) = 90^\circ + 15^\circ\)
\(2\alpha = 105^\circ\)
\(\alpha = \frac{105^\circ}{2}\)
\(\alpha = 52.5^\circ\)
To find \(\beta\), we can substitute the value of \(\alpha\) back into the first equation:
\(52.5^\circ + \beta = 90^\circ\)
\(\beta = 90^\circ - 52.5^\circ\)
\(\beta = 37.5^\circ\)
The two acute angles are \(52.5^\circ\) and \(37.5^\circ\). The question asks for one of the acute angles. Both are valid answers, but we need to choose from the given options.
Comparing with Options
The calculated acute angles are \(52.5^\circ\) and \(37.5^\circ\). Let's check the options provided:
| Option Number | Angle Measure |
|---|---|
| 1 | \(60^\circ\) |
| 2 | \(57.5^\circ\) |
| 3 | \(52.5^\circ\) |
| 4 | \(47.5^\circ\) |
Our calculated angle \(\alpha = 52.5^\circ\) matches option 3.
Conclusion
The two acute angles in the right-angled triangle are \(52.5^\circ\) and \(37.5^\circ\). One of these angles, \(52.5^\circ\), is listed as option 3.
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