Question

In a right triangle, the distance between the angle of 19 degrees is 1 mile, what is the opposite of this angle in feet with 4 decimal places

Answer 1

To find the length of the side opposite the angle of 19 degrees in a right triangle where the adjacent side is 1 mile, we can use the tangent function.

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Where:
- \(\theta\) is the angle (19 degrees in this case),
- opposite is the length of the side opposite the angle,
- adjacent is the length of the side adjacent to the angle (1 mile).

Given:
- \(\theta = 19\) degrees,
- adjacent = 1 mile.

We can rearrange the formula to solve for the opposite side:

\[
\text{opposite} = \tan(\theta) \times \text{adjacent}
\]

First, we need to convert 1 mile to feet. There are 5280 feet in a mile:

\[
\text{adjacent} = 1 \text{ mile} = 5280 \text{ feet}
\]

Next, we calculate the tangent of 19 degrees. Using a calculator:

\[
\tan(19^\circ) \approx 0.3443
\]

Now we can find the opposite side:

\[
\text{opposite} = \tan(19^\circ) \times 5280 \text{ feet} \approx 0.3443 \times 5280
\]

Calculating this gives:

\[
\text{opposite} \approx 1811.344
\]

Rounding to four decimal places:

\[
\text{opposite} \approx 1811.3440
\]

Thus, the length of the side opposite the 19-degree angle is approximately **1811.3440 feet**.