Question

find rs.
write your answer as an integer or as a decimal rounded to the nearest tenth.
rs =
submit

Explanation:

Step1: Identify trigonometric ratio

We have a right triangle $\triangle QSR$, right-angled at $Q$. For $\angle S = 23^\circ$, side $QS$ is adjacent, $QR = \sqrt{14}$ is opposite, and $RS$ is the hypotenuse. We use the tangent ratio first to find $QS$, then use Pythagoras theorem to find $RS$.
First, calculate $QS$:
$\tan(23^\circ) = \frac{QR}{QS}$
$QS = \frac{QR}{\tan(23^\circ)}$

Step2: Calculate length of QS

Substitute $QR = \sqrt{14} \approx 3.7417$, $\tan(23^\circ) \approx 0.4245$:
$QS \approx \frac{3.7417}{0.4245} \approx 8.814$

Step3: Apply Pythagoras theorem

$RS^2 = QS^2 + QR^2$
$RS = \sqrt{QS^2 + QR^2}$

Step4: Compute RS

Substitute $QS \approx 8.814$, $QR \approx 3.7417$:
$RS \approx \sqrt{(8.814)^2 + (3.7417)^2} \approx \sqrt{77.686 + 14.000} \approx \sqrt{91.686} \approx 9.6$

Alternatively, use sine ratio directly:

Step1: Use sine trigonometric ratio

$\sin(23^\circ) = \frac{QR}{RS}$
$RS = \frac{QR}{\sin(23^\circ)}$

Step2: Calculate RS

Substitute $QR = \sqrt{14} \approx 3.7417$, $\sin(23^\circ) \approx 0.3907$:
$RS \approx \frac{3.7417}{0.3907} \approx 9.6$

Answer:

$9.6$