Question

what is the measure of ∠r in △pqr? round to the nearest degree. 26° 42° 45° 64° (accompanied by a right - triangle diagram with right angle at p, side pr = 12.7, side qr = 14.1)

Explanation:

Step1: Identify triangle type and trigonometric ratio

$\triangle PQR$ is right - angled at $P$. We know the lengths of the opposite side ($PQ = 12.7$) and the hypotenuse ($QR=14.1$) with respect to $\angle R$. The sine ratio is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So, $\sin(\angle R)=\frac{PQ}{QR}=\frac{12.7}{14.1}$.

Step2: Calculate the value of $\sin(\angle R)$

$\frac{12.7}{14.1}\approx0.9007$.

Step3: Find $\angle R$

To find $\angle R$, we take the inverse sine (arcsin) of $0.9007$. So, $\angle R=\arcsin(0.9007)$. Using a calculator, $\arcsin(0.9007)\approx64^{\circ}$ (rounded to the nearest degree).

Answer:

$64^{\circ}$