Question

find the sine, cosine, and tangent of ∠r. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. sin(r) = cos(r) = tan(r) =

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ for an acute angle $\theta$. For $\angle R$, the opposite side to $\angle R$ is $PQ$, the adjacent side to $\angle R$ is $RP$, and the hypotenuse is $RQ$.

Step2: Identify side lengths

Given $RP = 12$, $RQ=20$, and using the Pythagorean theorem to find $PQ$. But we can directly use the given sides for trigonometric ratios. The opposite side $PQ=\sqrt{20^{2}-12^{2}}=\sqrt{(20 + 12)(20 - 12)}=\sqrt{32\times8}=\sqrt{256}=16$.

Step3: Calculate $\sin(R)$

$\sin(R)=\frac{PQ}{RQ}=\frac{16}{20}=\frac{4}{5}$

Step4: Calculate $\cos(R)$

$\cos(R)=\frac{RP}{RQ}=\frac{12}{20}=\frac{3}{5}$

Step5: Calculate $\tan(R)$

$\tan(R)=\frac{PQ}{RP}=\frac{16}{12}=\frac{4}{3}$

Answer:

$\sin(R)=\frac{4}{5}$
$\cos(R)=\frac{3}{5}$
$\tan(R)=\frac{4}{3}$