Knowing that sin 30 degrees = 1/2, what is a?

1. Identify the problem type: This is a trigonometry problem involving a right-angled triangle.
2. Analyze the given information: We are given a right-angled triangle with angles 30°, 60°, and 90°. The side adjacent to the 60° angle and opposite the 30° angle has a length of 9. The hypotenuse is labeled as 'a'. We are also given that $\sin 30^{\circ} = \frac{1}{2}$.
3. Determine the relevant trigonometric ratio: We need to find the length of the hypotenuse 'a'. We know the length of the side opposite the 30° angle (which is 9) and we want to find the hypotenuse. The sine function relates the opposite side and the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.
4. Set up the equation: Using the 30° angle, we have $\sin 30^{\circ} = \frac{9}{a}$.
5. Substitute the known value of sin 30°: We are given that $\sin 30^{\circ} = \frac{1}{2}$. So, the equation becomes $\frac{1}{2} = \frac{9}{a}$.
6. Solve for 'a': To solve for 'a', we can cross-multiply:
$1 \times a = 2 \times 9$
$a = 18$
Alternatively, we can use the 60° angle. The side opposite the 60° angle is 9, and the hypotenuse is 'a'. The sine of 60° is $\sin 60^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{a}$. We know that $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$. So, $\frac{\sqrt{3}}{2} = \frac{9}{a}$. Solving for 'a' gives $a = \frac{18}{\sqrt{3}} = \frac{18\sqrt{3}}{3} = 6\sqrt{3}$. This is not one of the options. Let's re-examine the diagram. The side with length 9 is opposite the 60° angle. The side 'a' is the hypotenuse. The side opposite the 30° angle is not given directly. However, in a 30-60-90 triangle, the sides are in the ratio $x : x\sqrt{3} : 2x$, where $x$ is the side opposite the 30° angle, $x\sqrt{3}$ is the side opposite the 60° angle, and $2x$ is the hypotenuse. Since the side opposite the 60° angle is 9, we have $x\sqrt{3} = 9$. Therefore, $x = \frac{9}{\sqrt{3}} = 3\sqrt{3}$. The hypotenuse 'a' is $2x$, so $a = 2(3\sqrt{3}) = 6\sqrt{3}$. This still does not match the options.

Let's reconsider the initial interpretation of the diagram. The side labeled 9 is adjacent to the 60° angle and opposite the 30° angle. The side labeled 'a' is the hypotenuse. The angle labeled 30° is opposite the side of length 9. The angle labeled 60° is opposite the side of unknown length.

Using the 30° angle:
The side opposite the 30° angle is 9.
The hypotenuse is 'a'.
The sine of 30° is the ratio of the opposite side to the hypotenuse.
$\sin 30^{\circ} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{a}$
We are given that $\sin 30^{\circ} = \frac{1}{2}$.
So, $\frac{1}{2} = \frac{9}{a}$.
Cross-multiplying gives:
$1 \times a = 2 \times 9$
$a = 18$.

Let's verify with the 60° angle.
The side adjacent to the 60° angle is 9.
The hypotenuse is 'a'.
The cosine of 60° is the ratio of the adjacent side to the hypotenuse.
$\cos 60^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{a}$
We know that $\cos 60^{\circ} = \frac{1}{2}$.
So, $\frac{1}{2} = \frac{9}{a}$.
Cross-multiplying gives:
$1 \times a = 2 \times 9$
$a = 18$.

Both methods yield the same result, confirming the answer.