Answer
$x=66.4$ cm
$y=60.2$ cm
$Y=42.2^{\circ}$
Work Step by Step
Step 1: To find $x$, we use the formula $\sin\theta=\frac{x}{z}$.
Step 2: $\sin47.8^{\circ}=\frac{x}{89.6}$
Step 3: $x=89.6\times\sin47.8^{\circ}$
Step 4: Using a calculator, $x\approx66.37$
Step 5: Rounding the answer to three significant degrees, $x\approx66.4$ cm.
Step 6: To find $y$, we use the formula $\cos\theta=\frac{y}{z}$.
Step 7: $\cos47.8^{\circ}=\frac{y}{89.6}$
Step 8: $y=89.6\times\cos47.8^{\circ}$
Step 9: Using a calculator, $y\approx60.19$
Step 10: Rounding the answer to three significant degrees, $y\approx60.2$ cm.
Step 11: As $X+Y=90^{\circ}$,
$Y=90^{\circ}-47.8^{\circ}$
Step 12: Solving, $Y=42.2^{\circ}$.