Question

what is the length of \\(\overline{ab}\\)? round to the nearest tenth.

triangle with right angle at c, ac = 10 m, angle at a is 75°, ab is labeled x, and multiple choice options: 9.7 m, 10.4 m, 37.3 m, 38.6 m

Explanation:

Step1: Identify trigonometric ratio

In right triangle \( \triangle ACB \), \( \angle C = 90^\circ \), \( AC = 10 \) m, \( \angle A = 75^\circ \). We use cosine: \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). Here, \( \cos(75^\circ)=\frac{AC}{AB} \), so \( AB = \frac{AC}{\cos(75^\circ)} \).

Step2: Calculate \( \cos(75^\circ) \)

\( \cos(75^\circ)\approx0.2588 \).

Step3: Compute \( AB \)

\( AB=\frac{10}{0.2588}\approx38.6 \) m.

Answer:

38.6 m (corresponding to the option with 38.6 m)