trigonometry angles of elevation and depression worksheet

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04-02-2025

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This worksheet will guide you through solving problems involving angles of elevation and depression using trigonometry. Understanding these concepts is crucial in various fields, from surveying and architecture to navigation and aviation. Let's dive in!

Understanding Angles of Elevation and Depression

Before we tackle problems, let's clarify the definitions:

• Angle of Elevation: This is the angle formed between the horizontal line of sight and the line of sight upward to an object. Imagine you're looking up at a bird; the angle from your eyes to the bird, measured from the horizontal, is the angle of elevation.

• Angle of Depression: This is the angle formed between the horizontal line of sight and the line of sight downward to an object. Think about looking down from a cliff at a boat; the angle from your eyes to the boat, measured from the horizontal, is the angle of depression.

Important Note: Angles of elevation and depression are always measured from the horizontal. They are also alternate interior angles which means they are equal.

Key Trigonometric Functions

Remember the three primary trigonometric functions (SOH CAH TOA):

• Sine (sin): Opposite / Hypotenuse
• Cosine (cos): Adjacent / Hypotenuse
• Tangent (tan): Opposite / Adjacent

These functions will be your tools for solving problems involving angles of elevation and depression. Always draw a diagram to visualize the problem; it makes identifying the opposite, adjacent, and hypotenuse sides much easier.

Practice Problems

Let's work through some examples. Remember to draw a diagram for each problem!

Problem 1: A surveyor stands 100 meters from the base of a building. The angle of elevation to the top of the building is 30°. How tall is the building?

Solution:

1. Draw a diagram: Draw a right-angled triangle with the surveyor's position at one corner, the base of the building at another, and the top of the building at the right angle.

2. Identify the known values: The adjacent side (distance from the surveyor to the building) is 100 meters, and the angle of elevation is 30°. We need to find the opposite side (height of the building).

3. Choose the appropriate trigonometric function: We have the adjacent side and need the opposite side, so we'll use the tangent function: tan(30°) = opposite / 100.

4. Solve for the unknown: opposite = 100 * tan(30°) ≈ 57.7 meters.

Therefore, the building is approximately 57.7 meters tall.

Problem 2: A hot air balloon is 500 meters above the ground. The angle of depression from the balloon to a landmark on the ground is 25°. How far is the landmark from the point on the ground directly below the balloon?

Solution:

1. Draw a diagram: Draw a right-angled triangle with the balloon at the top vertex, the point directly below the balloon on the ground at one corner, and the landmark at the other corner.

2. Identify the known values: The opposite side (height of the balloon) is 500 meters, and the angle of depression (which is equal to the angle of elevation from the landmark) is 25°. We need to find the adjacent side (distance to the landmark).

3. Choose the appropriate trigonometric function: We have the opposite side and need the adjacent side, so we'll use the tangent function: tan(25°) = 500 / adjacent.

4. Solve for the unknown: adjacent = 500 / tan(25°) ≈ 1072.3 meters.

Therefore, the landmark is approximately 1072.3 meters from the point on the ground directly below the balloon.

Further Practice

Try these problems on your own:

1. From the top of a lighthouse 50 meters high, the angle of depression to a boat is 15°. How far is the boat from the base of the lighthouse?

2. A ramp has an angle of elevation of 10°. If the ramp covers a horizontal distance of 20 meters, what is the height of the ramp?

Remember to always draw a diagram and carefully identify the known and unknown values before choosing the correct trigonometric function. Good luck!