Alright, let's break down the solution step-by-step to find the angle opposite the leg that is 6 inches long in this right triangle.
1. Identify the given information:
- One leg (let's call it [tex]\( a \)[/tex]) of the right triangle measures 6 inches.
- The other leg (let's call it [tex]\( b \)[/tex]) of the right triangle measures [tex]\( 6\sqrt{3} \)[/tex] inches.
2. Understand the problem:
- We need to find the measure of the angle opposite the leg that is 6 inches long.
3. Use the basic trigonometric ratio:
- In a right triangle, the tangent of an angle is the ratio of the length of the opposite leg to the length of the adjacent leg.
- So, for the angle [tex]\( \theta \)[/tex] opposite the leg measuring 6 inches:
[tex]\[
\tan(\theta) = \frac{\text{opposite leg}}{\text{adjacent leg}} = \frac{a}{b} = \frac{6}{6\sqrt{3}}
\][/tex]
4. Simplify the fraction:
- To simplify the fraction [tex]\( \frac{6}{6\sqrt{3}} \)[/tex], divide the numerator and the denominator by 6:
[tex]\[
\frac{6}{6\sqrt{3}} = \frac{1}{\sqrt{3}}
\][/tex]
- We can further simplify [tex]\( \frac{1}{\sqrt{3}} \)[/tex] by rationalizing the denominator:
[tex]\[
\frac{1}{\sqrt{3}} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3}
\][/tex]
5. Determine the angle:
- We now know that:
[tex]\[
\tan(\theta) = \frac{\sqrt{3}}{3}
\][/tex]
- In trigonometry, the angle [tex]\( \theta \)[/tex] whose tangent is [tex]\( \frac{\sqrt{3}}{3} \)[/tex] is [tex]\( 30^\circ \)[/tex].
6. Confirm the angle:
- The tangent function is well-defined and, for common angles, we know that:
[tex]\[
\tan(30^\circ) = \frac{\sqrt{3}}{3}
\][/tex]
- Therefore, the angle opposite the leg measuring 6 inches is indeed [tex]\( 30^\circ \)[/tex].
So, the measure of the angle opposite the leg that is 6 inches long is:
[tex]\[
\boxed{30^\circ}
\][/tex]
