Question

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

image of right triangle with right angle at c, ac = 10 m, angle at a is 75°, ab is labeled x

\\(\bigcirc\\) 9.7 m
\\(\bigcirc\\) 10.4 m
\\(\bigcirc\\) 37.3 m
\\(\bigcirc\\) 38.6 m

Explanation:

Step1: Identify adjacent side, hypotenuse

In right $\triangle ACB$, $\angle A=75^\circ$, adjacent side to $\angle A$ is $AC=10$ m, hypotenuse is $AB=x$.

Step2: Use cosine trigonometric ratio

Cosine of angle = $\frac{\text{adjacent}}{\text{hypotenuse}}$, so:
$\cos(75^\circ) = \frac{10}{x}$

Step3: Rearrange to solve for $x$

$x = \frac{10}{\cos(75^\circ)}$
Calculate $\cos(75^\circ)\approx0.2588$, so:
$x\approx\frac{10}{0.2588}\approx38.6$

Answer:

38.6 m