Question

question
in $delta rst$, $r = 8.4$ cm, $t = 8.1$ cm and $angle t=71^circ$. find all possible values of $angle r$, to the nearest 10th of a degree.

Explanation:

Step1: Apply Law of Sines

$\frac{\sin\angle R}{r} = \frac{\sin\angle T}{t}$

Step2: Substitute given values

$\frac{\sin\angle R}{8.4} = \frac{\sin71^\circ}{8.1}$

Step3: Isolate $\sin\angle R$

$\sin\angle R = \frac{8.4 \times \sin71^\circ}{8.1}$
Calculate $\sin71^\circ \approx 0.9455$, so:
$\sin\angle R = \frac{8.4 \times 0.9455}{8.1} \approx \frac{7.9422}{8.1} \approx 0.9805$

Step4: Find primary angle

$\angle R = \arcsin(0.9805) \approx 78.6^\circ$

Step5: Check for second possible angle

Since $\sin\theta = \sin(180^\circ-\theta)$, the second candidate is:
$180^\circ - 78.6^\circ = 101.4^\circ$
Verify if this is valid: Sum of angles would be $101.4^\circ + 71^\circ = 172.4^\circ < 180^\circ$, so it is valid.

Answer:

$78.6^\circ$ and $101.4^\circ$