Question

question 2 of 26
given the diagram below, what is \\(\sin(30^\circ)\\)?
triangle diagram: right triangle with 60° at top left, 30° at bottom right, hypotenuse 9, right angle at bottom left. text: triangle not drawn to scale
\\(\boldsymbol{\circ}\\) a. \\(\frac{1}{4}\\)
\\(\boldsymbol{\circ}\\) b. \\(\frac{3}{4}\\)
\\(\boldsymbol{\circ}\\) c. \\(\frac{1}{2}\\)
\\(\boldsymbol{\circ}\\) d. \\(\frac{1}{3}\\)

Explanation:

Step1: Recall sine definition in right triangle

In a right triangle, $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$. For $30^\circ$ angle, find opposite side.

Step2: Identify sides for $30^\circ$

In a 30 - 60 - 90 triangle, hypotenuse is 9. The side opposite $30^\circ$ is half the hypotenuse? Wait, no: wait, in 30 - 60 - 90 triangle, sides are in ratio $1:\sqrt{3}:2$ (opposite 30, 60, hypotenuse). Wait, hypotenuse is 9. Wait, no: the right triangle has angles 30°, 60°, 90°. So angle 30°: opposite side is the side opposite 30°, which is the shorter leg. Wait, hypotenuse is 9. Wait, no, wait: in a right triangle, for angle 30°, $\sin(30^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}$. Wait, but also, in a 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse? Wait, no, wait: hypotenuse is twice the shorter leg (opposite 30°). Wait, but maybe we can just recall that $\sin(30^\circ)=\frac{1}{2}$, regardless of the triangle's size? Wait, no, wait: wait, the triangle here has hypotenuse 9. Wait, no, wait: no, the sine of an angle in a right triangle is opposite over hypotenuse, but for 30°, the ratio is always 1/2, because in any 30 - 60 - 90 triangle, the side opposite 30° is half the hypotenuse. Wait, let's check: let the side opposite 30° be $x$, hypotenuse is 9. Then $\sin(30^\circ)=\frac{x}{9}$. But in a 30 - 60 - 90 triangle, $x=\frac{9}{2}$? Wait, no, wait: no, the ratio of sides in 30 - 60 - 90 is: side opposite 30° (shorter leg) = $a$, side opposite 60° (longer leg) = $a\sqrt{3}$, hypotenuse = $2a$. So if hypotenuse is 9, then $2a = 9\implies a=\frac{9}{2}$. Then $\sin(30^\circ)=\frac{a}{9}=\frac{\frac{9}{2}}{9}=\frac{1}{2}$. So regardless of the hypotenuse length, $\sin(30^\circ)=\frac{1}{2}$. Wait, but that's a standard angle. So $\sin(30^\circ)=\frac{1}{2}$.

Answer:

C. $\frac{1}{2}$