Question

ac = \boxed{} round your answer to the nearest hundredth. diagram: right triangle with right angle at c, angle at a is 65°, segment cb = 2, segment ac (marked ?), vertices a, b, c

Explanation:

Step1: Identify the trigonometric relationship

In the right triangle \( \triangle ABC \) (right - angled at \( C \)), we know the length of the side opposite the \( 65^{\circ} \) angle (\( BC = 2 \)) and we want to find the length of the adjacent side (\( AC \)) to the \( 65^{\circ} \) angle. The tangent function relates the opposite and adjacent sides of a right triangle: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 65^{\circ} \), opposite side \( = BC = 2 \), and adjacent side \( = AC \). So, \( \tan(65^{\circ})=\frac{BC}{AC} \).

Step2: Solve for \( AC \)

From \( \tan(65^{\circ})=\frac{2}{AC} \), we can re - arrange the formula to solve for \( AC \). Cross - multiplying gives us \( AC\times\tan(65^{\circ}) = 2 \), then \( AC=\frac{2}{\tan(65^{\circ})} \). We know that \( \tan(65^{\circ})\approx2.1445 \). Substituting this value into the formula, we get \( AC=\frac{2}{2.1445}\approx0.93 \).

Answer:

\( 0.93 \)