Question

which equation can be used to find the length of \\(overline{ac}\\)? \\(\frac{10}{cos(40^circ)} = ac\\) \\(\frac{10}{sin(40^circ)} = ac\\) \\((10)cos(40^circ) = ac\\) \\((10)sin(40^circ) = ac\\)

Explanation:

Step1: Recall SOHCAHTOA

In a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Step2: Identify sides for $\angle B$

For $\angle B = 40^\circ$, the hypotenuse is $AB = 10$ ft, and the side opposite to $\angle B$ is $AC$ (since $\angle C$ is right angle, $AC$ is opposite $\angle B$).

Step3: Apply sine formula

Using $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, substitute $\theta = 40^\circ$, opposite = $AC$, hypotenuse = $10$. So $\sin(40^\circ) = \frac{AC}{10}$.

Step4: Solve for $AC$

Multiply both sides by $10$: $10 \cdot \sin(40^\circ) = AC$.

Answer:

$\boldsymbol{(10)\sin(40^\circ) = AC}$ (the option with this equation)