Question

A boat is tied to the tip of a post on a dock. The rope and the post form a right triangle with the rope as the hypotenuse. The rope is 60 inches long and the post is 3 feet tall. Using the inverse of sine, what is the approximate angle formed between the rope and the post?

87 degrees
37 degrees
48 degrees
53 degrees

Answer 1

To find the angle between the rope and the post using the inverse sine function, we first need to convert all measurements to the same unit. The height of the post is given as 3 feet, which we convert to inches:

\[
3 \text{ feet} = 3 \times 12 \text{ inches} = 36 \text{ inches}
\]

Now we have a right triangle where:
- The height of the post (opposite side) is 36 inches.
- The length of the rope (hypotenuse) is 60 inches.

We can use the sine function, which is defined as:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Substituting in the values we have:

\[
\sin(\theta) = \frac{36 \text{ inches}}{60 \text{ inches}} = \frac{36}{60} = 0.6
\]

Now, we can find the angle \(\theta\) by taking the inverse sine:

\[
\theta = \sin^{-1}(0.6)
\]

Using a calculator to find this value:

\[
\theta \approx 36.87^{\circ}
\]

For simplicity, we can round this to approximately:

\[
\theta \approx 37^{\circ}
\]

Thus, the approximate angle formed between the rope and the post is:

\[
\boxed{37 \text{ degrees}}
\]