Problem 19 - Algebra 2 and Trigonometry

Question

A telephone pole on a hillside makes an angle of 78 degrees with the upward slope. A wire from the top of the pole to a point up the hill is 12.0 feet long and makes an angle of 15 degrees with the pole. a. Find, to the nearest hundredth, the distance from the foot of the pole to the point at which the wire is fastened to the ground. b. Use the answer to part a to find, to the nearest tenth, the height of the pole.

Step-by-Step Solution

Answer

a. 11.59 feet b. 3.1 feet

Step 1: Understanding the Triangle

To solve this problem, we consider a triangle formed by the pole, the ground, and the wire. The wire forms the hypotenuse of this triangle with a length of 12 feet. The angle between the wire and the pole is given as 15 degrees, which will be used to find the other dimensions of the triangle.

Step 2: Calculate the Distance Along the Slope

We use the cosine rule or the definition of cosine in a right triangle to calculate the distance from the foot of the pole to the point where the wire is fastened on the ground. This distance forms the adjacent side relative to the 15-degree angle. We use the formula:\[\cos(15^{\circ}) = \frac{\text{adjacent side (distance along slope)}}{12}\]Solving for the adjacent side, we get:\[\text{distance along slope} = 12 \times \cos(15^{\circ})\]Calculating this gives approximately 11.59 feet.

Step 3: Calculate the Height of the Pole

The height of the pole forms the opposite side in our triangle for the 15-degree angle. We use the sine rule to calculate the height using the formula:\[\sin(15^{\circ}) = \frac{\text{opposite side (height of pole)}}{12}\]Solving for the opposite side, we find:\[\text{height of the pole} = 12 \times \sin(15^{\circ})\]Calculating this gives approximately 3.1 feet.

Key Concepts

Triangle Properties

One of the basics of trigonometry is understanding how triangles work. Triangles are geometric shapes with three sides and three angles. In our problem with the telephone pole on a hillside, we are faced with a specific type of triangle. This is primarily due to the angle relationships and sides involved, forming a perfect scenario to explore their properties.

Key properties of triangles include:

• Every triangle has three sides, three angles, and the sum of its angles is always 180 degrees.
• Triangles can be classified based on their angles: acute (all angles less than 90 degrees), obtuse (one angle more than 90 degrees), or right (one angle exactly 90 degrees).
• The relationships between the angles and sides of a triangle are crucial for solving problems.

By understanding these properties, we can effectively use trigonometric functions to find unknown sides and angles. For our problem, knowing that the wire is the hypotenuse allows us to use right triangle trigonometric principles.

Angle Calculations

Angles are a fundamental aspect of trigonometry and triangles. In this context, the angle calculations are central to determining unknown dimensions within the triangle.

For the telephone pole problem, the known angles are 78 degrees with the slope and 15 degrees between the wire and the pole. Calculating angles accurately can be crucial, as they directly affect the lengths of sides through trigonometric functions like sine and cosine.

We must carefully use these angles to set up our equations. For example, by knowing the angle of 15 degrees as the angle between the hypotenuse and the adjacent side, we can apply the cosine function to find the adjacent side, which is the distance from the pole's base to where the wire is fastened on the ground.

Trigonometric Functions

Trigonometric functions are mathematical tools used to relate the angles of a triangle to its sides. The primary trigonometric functions—sine, cosine, and tangent—are particularly useful when dealing with right triangles.

Here's how they help in solving the telephone pole problem:

• The **cosine** function relates the adjacent side and hypotenuse: \[\cos(15^{\circ}) = \frac{\text{adjacent}}{\text{hypotenuse}}\]
• The **sine** function relates the opposite side and hypotenuse:\[\sin(15^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}}\]

By applying these functions, we can solve for the length of the adjacent side (distance along the slope) and the opposite side (height of the pole). This illustrates the power of trigonometric functions in solving practical geometry problems.

Right Angle Triangle

Right angle triangles are a special type of triangle that contains one 90-degree angle. This property makes them incredibly important in trigonometry, as they simplify many calculations.
In the problem with the telephone pole, the triangle in question is a right triangle. This is because the wire and the ground can be thought of forming a right angle where the wire meets the slope vertically.

• The side opposite the 90-degree angle is called the hypotenuse, which in this case, is the wire.
• The other two sides, referred to as the opposite and adjacent sides, are aligned with the angle of interest, which is 15 degrees here.
• The relationships in a right triangle help in using functions like sine and cosine to find side lengths as shown in the steps of our solution.

Understanding how right triangles work allows us to use these trigonometric identities efficiently to solve the original problem, ensuring that each calculation leads to accurate results.