Question

the equation \\(\sin(25^{\circ}) = \frac{9}{c}\\) can be used to find the length of \\(\overline{ab}\\). what is the length of \\(\overline{ab}\\)? round to the nearest tenth.
options:
\\(\circ\\) 19.3 in.
\\(\circ\\) 21.3 in.
\\(\circ\\) 23.5 in.
\\(\circ\\) 68.0 in.
(there is a right triangle with right angle at c, angle at a is 25°, side bc is 9 in, and hypotenuse ab is labeled as c.)

Explanation:

Step1: Identify the triangle type

The triangle is right - angled at \(C\), with \(\angle A = 25^{\circ}\) and the side opposite to \(\angle A\) (i.e., \(BC\)) is \(9\) in. We need to find the length of the hypotenuse \(AB\) (denoted as \(c\)).
From the definition of sine in a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 25^{\circ}\), opposite side \(= 9\) in, and hypotenuse \(= c\) (length of \(AB\)). So, \(\sin(25^{\circ})=\frac{9}{c}\).

Step2: Solve for \(c\)

We can re - arrange the formula \(\sin(25^{\circ})=\frac{9}{c}\) to solve for \(c\). Cross - multiplying gives us \(c\times\sin(25^{\circ}) = 9\), then \(c=\frac{9}{\sin(25^{\circ})}\).
We know that \(\sin(25^{\circ})\approx0.4226\).
So, \(c=\frac{9}{0.4226}\approx21.3\) in.

Answer:

21.3 in (corresponding to the option "21.3 in")