Question

which equation can be used to find the length of $overline{ac}$?
image of a right - triangle abc with right - angle at c, ab = 10 in., angle b = 40 degrees
(10)sin(40°)=ac
(10)cos(40°)=ac
$\frac{10}{sin(40°)} = ac$
$\frac{10}{cos(40°)} = ac$

Explanation:

Step1: Recall sine - cosine definitions in right - triangle

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. In right - triangle $ABC$ with right angle at $C$, $\angle B = 40^{\circ}$ and hypotenuse $AB = 10$ inches. We want to find the length of $AC$ which is opposite to $\angle B$.

Step2: Apply the sine formula

Since $\sin B=\frac{AC}{AB}$, and $AB = 10$ and $B = 40^{\circ}$, we can rewrite the formula as $AC=AB\times\sin B$. Substituting the values, we get $AC = 10\times\sin(40^{\circ})$.

Answer:

$(10)\sin(40^{\circ}) = AC$