Question

question 1 of 10
the side lengths of a 30-60-90 triangle are in the ratio $1 : \sqrt{3} : 2$. what is $\sin 30^\circ$?

a. $\frac{\sqrt{3}}{3}$
b. $\sqrt{3}$
c. $\frac{1}{2}$
d. $\frac{\sqrt{3}}{2}$

Explanation:

Step1: Recall Sine Definition

Sine of an angle in a right triangle is $\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Identify 30° Triangle Sides

In a 30 - 60 - 90 triangle, the side opposite 30° is the shortest side (length 1), hypotenuse is 2 (from ratio $1:\sqrt{3}:2$).

Step3: Calculate $\sin30^{\circ}$

Using $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, for $\theta = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$. Wait, but looking at the options (assuming option C is $\frac{1}{2}$ and maybe a typo in display, or maybe the options were misrendered. Wait, re - checking: the ratio is $1:\sqrt{3}:2$ (opposite 30°: opposite 60°: hypotenuse). So $\sin30^{\circ}=\frac{\text{opposite to }30^{\circ}}{\text{hypotenuse}}=\frac{1}{2}$. If option C is $\frac{1}{2}$ (assuming the display for C is $\frac{1}{2}$), then that's the answer.

Answer:

C. $\frac{1}{2}$ (assuming the option C is $\frac{1}{2}$ as per the 30 - 60 - 90 triangle sine calculation)