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To solve for angle \( \alpha \) in this right triangle, we can use trigonometric ratios. Since we are given the lengths of the opposite side \( a = 3 \) and the adjacent side \( b = 7 \), we will use the tangent function:

\[
\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{3}{7}
\]

Now, to find \( \alpha \), we take the arctangent (inverse tangent) of the ratio:

\[
\alpha = \tan^{-1} \left( \frac{3}{7} \right)
\]

Let me calculate that for you.The measure of angle \( \alpha \) is approximately \( 23.2^\circ \) when rounded to the nearest tenth.

Would you like further details on this solution, or do you have any questions?

Here are some related questions for further practice:
1. What is the value of angle \( \beta \) in this triangle?
2. How would you calculate the area of the triangle?
3. Can you use the sine or cosine functions to verify \( \alpha \)?
4. How would the values change if \( a \) were doubled while \( b \) stayed the same?
5. What is the relationship between the angles in a right triangle?

**Tip:** Always check your results by ensuring the angles in a triangle add up to 180°.

Solve the right triangle where a = 3 and b = 7. Find the length of side c and the measure of angle α.

Learn how to solve a right triangle with sides a = 3 and b = 7. This solution covers finding the length of the hypotenuse using the Pythagorean theorem and calculating the angle α using the inverse tangent function. Suitable for high school trigonometry students.

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