Problem 49 - Precalculus with Limits

Question

Angle of Elevation A 10 -meter utility pole casts a 17 -meter shadow directly down a slope when the angle of elevation of the sun is \(42^{\circ}\) (see figure). Find \(\theta,\) the angle of elevation of the ground.

Step-by-Step Solution

Answer

The angle of elevation of the ground can be found using the above steps, using the given data, applying the tangent of the triangle principle and calculating the unknown angle using inverse tangent.

Step 1: Determining the right triangle

From the information given in the problem, a right triangle is formed with the utility pole as one side, the slope as the hypotenuse and the shadow of the pole as other side. The angle of elevation of the sun is \(42 ^\circ\) and this angle is formed by the slope and the shadow of the pole.

Step 2: Using tangent of angle

In a right triangle, the tangent of an angle is given by the ratio of the length of the side opposite to the angle to the length of the side adjacent to it. So, the equation is \[\tan(42^{\circ}) = \frac{10}{17}.\] By cross-multiplication, we get \(10 \cdot \tan(42^{\circ}) = 17.\]

Step 3: Solving for the slope

Now, in order to find out the angle θ, the equation can be rewritten in terms of θ as \[\tan(\theta) = \frac{17}{10 \cdot \tan(42^{\circ})}.\] Now, calculate \(\theta\) by finding the inverse tangent (also known as arctan) of the right side. Hence, \(\theta = \arctan \left( \frac{17}{10 \cdot \tan(42^{\circ})} \right).\]

Step 4: Calculate the angle

Now, using a calculator, the value of \(\theta\) can be calculated by solving the above equation. Therefore, the angle θ is the angle of elevation of the ground.

Key Concepts

Angle of Elevation

The angle of elevation is an important concept in trigonometry. It helps describe a type of visual angle. Imagine looking up at an object above the horizontal line from your eye level. This angle between the line of sight and the horizontal line is your angle of elevation. It is an 'upward' angle.
In real-life scenarios, this concept helps in various tasks, such as determining heights of objects like buildings or trees.
When you know certain measurements, like the distance from the object and the angle of elevation, you can calculate how tall something is. This is very useful, for example, when you don't have direct access to the object.

• **Remember**: The angle of elevation is always measured from the horizontal up.
• **Practical Tip**: Standing at a certain distance from an object and using this angle helps in creating right triangles for calculation purposes.

Right Triangle

A right triangle is a triangle that has one angle equal to 90 degrees. This simple geometric shape is a core part of trigonometry because it allows us to apply various trigonometric functions.
The right angle defines the relationships between the triangle's sides: the hypotenuse, opposite side, and adjacent side. These terms are critical in using functions such as sine, cosine, and tangent.
In our example, the scenario helps form a right triangle with:

• One side being the height of the utility pole.
• The hypotenuse as the slope.
• The shadow of the pole forms the adjacent side.

This structure of a right triangle allows the application of the tangent function, where specific angles and side lengths can be calculated.

Tangent Function

The tangent function is one of the primary functions used in trigonometry. It relates the angles of a right triangle to the ratio of two specific sides: \( an( heta) = \frac{\text{opposite}}{\text{adjacent}}\).
In our case, the angle is the given angle of the sun, and the sides are the height of the pole (opposite) and the shadow on the ground (adjacent).
What's neat about the tangent function is how it offers a straightforward way to connect an angle with a simple ratio. This simplification is powerful in problems where only a couple of measurements are known.

• **Key Use**: Helps solve for unknown angles or sides of the triangle when one side length and one angle is known.
• **Example Setup**: Tangent is especially useful in settings like slopes or inclines where measuring direct vertical or horizontal distances is challenging.

Inverse Tangent

Inverse tangent, also known as arctangent, is essentially the reverse operation of the tangent function. It helps us find an angle when we know a ratio of the sides. This is particularly useful when the goal is to find the actual angle from a known ratio.
Mathematically, it is expressed as \( heta = \arctan(k)\), where \(k\) is the ratio between the opposite and adjacent sides. In our example, once we determine the tangent ratio between the slope and the known side lengths, using the inverse tangent function will calculate the angle θ.

• **Usage**: Essential in problems where the angle needs to be derived from given side measurements.
• **Practical Tip**: This function is often readily available on scientific calculators by pressing the "tan")." button.

Understanding and mastering both tangent and inverse tangent functions become pivotal when solving trigonometric problems involving right triangles.