Question

solve for x. round to the nearest tenth, if necessary.
triangle with right angle at y, side ty = 77, angle at z is 68 degrees, hypotenuse tz = x

Explanation:

Step1: Identify trigonometric ratio

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

Step2: Rearrange to solve for $x$

Isolate $x$ by cross-multiplying:
$x=\frac{77}{\sin(68^\circ)}$

Step3: Calculate the value

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

Answer:

$83.0$