Since $\sin R=\frac{5}{13}$, the side opposite angle $R$ is 5 and the hypotenuse is 13. In a right triangle with side lengths in that ratio, the missing leg is 12 because $5^2+12^2=13^2$. Tangent is opposite over adjacent, so $\tan R=\frac{5}{12}$.

In a right triangle, $\sin (\theta )$ is defined as

The side opposite $\theta$ is 6 and the hypotenuse is 10, so

This simplifies to

So the correct answer is $\sin (\theta )=\frac{6}{10}=\frac{3}{5}$.

The cable is the hypotenuse of the right triangle, and the height of the pole is opposite the $37^{\circ }$ angle because that angle is formed between the cable and the ground. Using sine,

So,

Therefore, the correct expression is $10\sin 37^{\circ }$.

Because the ladder, wall, and ground form a right triangle, the ladder is the hypotenuse. The side along the ground is 6, and the hypotenuse is 10. First find the height using the Pythagorean theorem: $6^2+h^2=10^2$, so $36+h^2=100$, which gives $h^2=64$ and $h=8$. For angle $\theta$, formed by the ladder and the ground, the opposite side is the height $8$ and the hypotenuse is $10$. Therefore, $\sin \theta =\frac{8}{10}=\frac{4}{5}$.

The ladder, wall, and ground form a right triangle. The ladder is the hypotenuse, and the distance from the wall to the base is the side adjacent to the $60^{\circ }$ angle. Use cosine:

$\cos 60^{\circ }=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{8}{L}$

Since $\cos 60^{\circ }=\frac{1}{2}$,

$\frac{1}{2}=\frac{8}{L}$

Multiply both sides by $L$:

$\frac{L}{2}=8$

So $L=16$.

Therefore, the ladder is 16 feet long.