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Applications of Right Triangle Trigonometry
Introduction: In this lesson, trigonometric ratios (sine, cosine, and tangent) will be used in real world applications. First we will review the basics: In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.
We shall call the opposite side "opp," the adjacent side "adj" and the hypotenuse "hyp."
Definitions: In the following definitions, sine is called "sin," cosine is called "cos" and tangent is called "tan." The origin of these terms relates to arcs and tangents to a circle.
  1. sin(A) =
  2. cos(A) =
  3. tan(A) =
Special Angles:

Let's Practice:
  1. A tree casts a 60 foot shadow. The angle of elevation is 30º. This is the angle at which you look up to the top of the tree from the ground. What is the height of the tree?

We have .
Therefore feet.
  1. A brick pathway is 30 yards long. A square courtyard is being built incorporating this pathway diagonally. How long should each side of the square be?
A diagram showing the square and the required pathway is shown below.

We have cos(45º) = .  Therefore yards.
  1. Looking down from the roof of a house at an angle of 23º a shiny object is seen. This 23º angle is with respect to the horizontal (see diagram below). The roof of the house is 32 feet above the ground. How far is the shiny object from the house?

Looking down from the top of the house, the 23º angle is called the angle of depression. Notice by alternate interior angles that the angle from the ground to the house is also 23º. From this diagram we can see that the". Since the tangent of 23º is approximately 0.42447 (check this value with a calculator) we can set up the following equation:

Knowing that the house roof is 32 feet above the ground let's us write the equation = 32 giving us feet, the distance from the house to the shiny object.
  1. A lamp post is 10 feet tall and casts a shadow. The distance from the top of the lamp post to the tip of the shadow is 20 feet. What is the angle of elevation shown as angle A in the diagram shown below?

10 feet A
shadow = 20 feet

We have tan(A) = opp/adj = 10/20 = 1/2. This is not one of our special angles."

This is not one of the special angles. To get the correct value for A we need to use a calculator's inverse tangent feature, tan-1. On a TI-83 calculator we use We recognize this as the tan(26.5651º). Therefore angle A is 26.5651º.

Practicing further, what would angle A be if the lamp post had only been 7 feet tall?

We would then have tan(A) = opp/adj = 7/20 = 0.35

Or, A = tan-1(0.35). This tells us that angle A is approximately 20.4873º.

Example An observer on top of a 60-foot tall lighthouse sees a boat in distress at a 5º angle of depression. How far is the boat from the base of the lighthouse?
What is your answer?
Example A square has a diagonal of 20 feet. What is the area of this square?
What is your answer?
Example A tree casts a shadow 70 feet long at an angle of elevation of 30º. How tall is the tree?
What is your answer?
Example You are looking up at a fourth story window, 40 feet up in a building. You are 100 feet away from the building, across the street. What is the angle of elevation?
What is your answer?

M Ransom

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