Question

solve for $x$. round to the nearest tenth, if necessary.
image of a right triangle tyz with right angle at y, ty = 77, angle at z is 68 degrees, and hypotenuse tz = x

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle, angle at Z is $68^\circ$, adjacent side to this angle is $YZ$, opposite side is $TY=77$, and $x$ is the hypotenuse. Use sine: $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$

Step2: Substitute known values

$\sin(68^\circ)=\frac{77}{x}$

Step3: Rearrange to solve for $x$

$x=\frac{77}{\sin(68^\circ)}$

Step4: Calculate value

First, $\sin(68^\circ)\approx0.9272$, so $x\approx\frac{77}{0.9272}\approx83.0$

Answer:

$83.0$