Question

At \(10 \mathrm{AM}\) on April 26,2018 , a building 300 feet high cast a shadow 50 feet long. What was the angle of elevation of the Sun?

Short answer

The angle of elevation of the Sun is approximately \(80.54^\text{°}\).

Step-by-step solution

Identify the Known Values

The height of the building is 300 feet. The length of the shadow is 50 feet.

Draw a Diagram

Sketch a right triangle where the height of the building represents the opposite side and the length of the shadow represents the adjacent side.

Use the Tangent Function

In a right triangle, the tangent of the angle of elevation \(\theta\) is the ratio of the opposite side to the adjacent side: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

Plug in the Known Values

Using the known values: \(\tan(\theta) = \frac{300}{50}\). Simplify this to get \(\tan(\theta) = 6\).

Calculate the Angle

Use the arctangent (inverse tangent) function to find the angle: \(\theta = \tan^{-1}(6)\).

Find the Angle Using a Calculator

Using a calculator, \(\theta = \tan^{-1}(6) \approx 80.54^\text{°} \).

Key concepts

tangent function

In trigonometry, the tangent function is a key concept that helps relate the angles of a right triangle to its side lengths. The tangent of an angle \(\theta\) in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This can be expressed as:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]
In the given problem, you need to find the angle of elevation of the Sun using the height of the building (opposite side) and the shadow's length (adjacent side). Since you know these lengths, you can easily use the tangent function to solve for \(\theta\).

inverse tangent

To find the angle when you know the tangent value, you use the inverse tangent function (also known as arctangent), denoted as \(\tan^{-1}(x)\). In this problem, once you've calculated \(\tan(\theta) = 6\), the next step is to find the angle \(\theta\).
The formula is:
\[\theta = \tan^{-1}(6)\]
Using a scientific calculator or a calculator app, you input the value to obtain \(\theta\). For this problem, it approximately equals 80.54 degrees. This means the angle of elevation of the Sun is roughly 80.54 degrees.

trigonometry

Trigonometry deals with the relationships between angles and sides of triangles, especially right-angled triangles. It introduces functions like sine, cosine, and tangent, which are fundamental for solving real-world problems involving right triangles. In this problem, we used the tangent function.
Key trigonometric functions:

• Sine: \(\frac{\text{opposite}}{\text{hypotenuse}}\)
• Cosine: \(\frac{\text{adjacent}}{\text{hypotenuse}}\)
• Tangent: \(\frac{\text{opposite}}{\text{adjacent}}\)

Trigonometry is not only useful in math but also in fields such as physics, engineering, and astronomy.

right triangle

A right triangle is a triangle with one angle measuring 90 degrees. It consists of three sides: the hypotenuse (the longest side opposite the right angle), the opposite side (opposite the angle you're interested in), and the adjacent side (next to the angle you're interested in).
In the current exercise, understanding the setup of a right triangle simplifies the problem. The height of the building forms the opposite side, the shadow length represents the adjacent side, and these fit perfectly into the right triangle model to measure the Sun's angle of elevation.
Remember, in any right-angled triangle problem, identifying these sides correctly helps apply the trigonometric functions accurately.