To find the angle of elevation of the sun when a 12.5-meter tall telephone pole casts an 18-meter long shadow, follow these steps:
1. Understand the problem and set up the situation:
- We have a telephone pole that is 12.5 meters tall (this will be our "opposite" side in a right triangle).
- The shadow cast by the pole is 18 meters long (this will be our "adjacent" side in the same right triangle).
2. Identify the trigonometric function to use:
- In this scenario, we are dealing with a right triangle where we know the lengths of the opposite and adjacent sides. To find the angle of elevation, we can use the tangent function because [tex]\(\tan(\theta)\)[/tex] is the ratio of the opposite side to the adjacent side.
- The tangent of an angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Set up the equation:
- Substitute the known values into the tangent equation:
[tex]\[
\tan(\theta) = \frac{12.5}{18}
\][/tex]
4. Solve for [tex]\(\theta\)[/tex]:
- To find the angle [tex]\(\theta\)[/tex], we need to use the arctangent (inverse tangent) function, which is often denoted as [tex]\(\tan^{-1}\)[/tex] or [tex]\(\text{arctan}\)[/tex]:
[tex]\[
\theta = \tan^{-1}\left(\frac{12.5}{18}\right)
\][/tex]
5. Convert the angle from radians to degrees:
- The arctangent function will give us the angle in radians. To convert radians to degrees, recall that:
[tex]\[
\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right)
\][/tex]
The calculation gives us two values:
- The angle of elevation in radians: approximately 0.607 radians.
- The angle of elevation in degrees: approximately 34.78 degrees.
Thus, the angle of elevation of the sun is about 0.607 radians or 34.78 degrees.
