Question

1. use the information in the table and image to find the side - length ac. a. $\frac{sqrt{2}}{2}$ b. $\frac{sqrt{4}}{2}$ c. 2 d. $sqrt{2}$

Explanation:

Step1: Recall sine - cosine - tangent relations

In right - triangle ABC, if we want to find the length of side AC, and we know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\angle A = 45^{\circ}$, and $AB = 1$. We can use the cosine function since $\cos A=\frac{AB}{AC}$.

Step2: Substitute values

We know that $\cos45^{\circ}=\frac{\sqrt{2}}{2}$, and $\cos A=\frac{AB}{AC}$. Substituting $A = 45^{\circ}$ and $AB = 1$ into the formula $\cos A=\frac{AB}{AC}$, we get $\frac{\sqrt{2}}{2}=\frac{1}{AC}$.

Step3: Solve for AC

Cross - multiply to get $AC\times\sqrt{2}=2$, then $AC = \sqrt{2}$.

Answer:

D. $\sqrt{2}$