Question

find the length of side x.
in a right triangle, one side is 23, one acute angle is 63°, find the length of side x (the side opposite the right angles adjacent angle of 63°)
options:
61.0
50.7
50.1
60.7

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle, with the side opposite the 63° angle equal to 23, and $x$ is the hypotenuse. We use the sine function: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$

Step2: Rearrange to solve for x

Rearrange the formula to isolate $x$: $x=\frac{\text{opposite}}{\sin(\theta)}$

Step3: Substitute values and calculate

Substitute $\theta=63^\circ$, opposite=23:
$x=\frac{23}{\sin(63^\circ)}$
Calculate $\sin(63^\circ)\approx0.8910$, so $x\approx\frac{23}{0.8910}\approx25.81$
Wait, correction: Re-identify sides: The given side (23) is adjacent to the 63° angle, $x$ is the hypotenuse. Use cosine: $\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$
Rearrange: $x=\frac{\text{adjacent}}{\cos(\theta)}$
Substitute values: $x=\frac{23}{\cos(63^\circ)}$
$\cos(63^\circ)\approx0.4540$, so $x\approx\frac{23}{0.4540}\approx50.66\approx50.7$

Answer:

50.7