Question

triangle abc is shown. what is sin(∠abc)? choose 1 answer: a √2 b 1 c 3√2 d √2/2

Explanation:

Step1: Recall sine - function definition

The sine of an angle in a right - triangle is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. In right - triangle $ABC$, for $\angle ABC$, the opposite side to $\angle ABC$ is $AC = 3$ and the hypotenuse is $AB=3\sqrt{2}$.

Step2: Calculate $\sin(\angle ABC)$

Using the formula $\sin(\angle ABC)=\frac{AC}{AB}$, substitute $AC = 3$ and $AB = 3\sqrt{2}$ into it. We get $\sin(\angle ABC)=\frac{3}{3\sqrt{2}}$. Simplify the fraction: $\frac{3}{3\sqrt{2}}=\frac{1}{\sqrt{2}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$, so $\frac{1}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Answer:

D. $\frac{\sqrt{2}}{2}$