find the sine, cosine, and tangent of ∠r.\nsimplify your answers and write them as proper fractions…

Explanation:

Step1: Find the length of the other side

Use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$. Let the unknown side be $x$. We know $c = 65$ and one side $b = 56$. Then $x=\sqrt{65^{2}-56^{2}}=\sqrt{(65 + 56)(65 - 56)}=\sqrt{121\times9}=\sqrt{1089}=33$.

Step2: Calculate $\sin(R)$

The sine of an angle in a right - triangle is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle R$, the opposite side to $\angle R$ is $ST = 56$ and the hypotenuse is $RS=65$. So $\sin(R)=\frac{56}{65}$.

Step3: Calculate $\cos(R)$

The cosine of an angle in a right - triangle is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\angle R$, the adjacent side to $\angle R$ is $RT = 33$ and the hypotenuse is $RS = 65$. So $\cos(R)=\frac{33}{65}$.

Step4: Calculate $\tan(R)$

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle R$, the opposite side is $ST = 56$ and the adjacent side is $RT = 33$. So $\tan(R)=\frac{56}{33}$.

Answer:

$\sin(R)=\frac{56}{65}$, $\cos(R)=\frac{33}{65}$, $\tan(R)=\frac{56}{33}$