Heights and Distances: Angle of Elevation MCQs Quiz | Class 10

Understanding Heights and Distances: Angle of Elevation

Heights and Distances is a practical application of trigonometry that helps us find the heights of objects and distances between them without direct measurement. It primarily uses trigonometric ratios and the concepts of angles of elevation and depression.

Key Concepts: Angle of Elevation

When an observer looks at an object that is above their eye level, the angle formed between the line of sight and the horizontal line is called the Angle of Elevation.

• Line of Sight: This is the imaginary line drawn from the eye of the observer to the point on the object that the observer is viewing.
• Horizontal Line: This is the imaginary line passing through the eye of the observer, parallel to the ground.
• Angle of Elevation: It is always measured upwards from the horizontal line.

Diagram Interpretation: Imagine a person standing on the ground looking up at the top of a tower. The line from the person’s eye to the top of the tower is the line of sight. The line from the person’s eye parallel to the ground is the horizontal line. The angle between these two lines is the angle of elevation.

Trigonometric Ratios (Quick Recap)

In a right-angled triangle:

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

For problems involving Angle of Elevation, the tangent ratio (tan) is most frequently used because it relates the height (opposite side) and the distance from the base (adjacent side) to the angle of elevation.

Steps to Solve Problems

1. Draw a Diagram: Always start by drawing a clear, labeled diagram that represents the given situation. This helps in visualizing the right-angled triangle(s).
2. Identify Knowns and Unknowns: Mark all given measurements (angles, heights, distances) and the quantity you need to find.
3. Choose the Right Ratio: Based on the known and unknown sides/angles, select the appropriate trigonometric ratio (sin, cos, or tan). For angle of elevation, if height and base distance are involved, tan is usually the choice.
4. Formulate Equation: Set up the trigonometric equation (e.g., tan(angle) = height / distance).
5. Solve: Solve the equation to find the unknown value. Remember to use the correct values for trigonometric functions of standard angles.

Common Tangent Values

Quick Revision Checklist

• Angle of Elevation is always measured upwards from the horizontal.
• The line of sight connects the observer’s eye to the object.
• Horizontal line is parallel to the ground.
• Most problems use the tangent ratio (tan = Opposite/Adjacent).
• Always draw a diagram to simplify the problem.

Practice Questions

Try solving these additional problems to solidify your understanding:

1. A tower stands vertically on the ground. From a point on the ground, which is 15 meters away from the foot of the tower, the angle of elevation of the top of the tower is 60 degrees. Find the height of the tower.
2. A pole 6 meters high casts a shadow 2 sqrt(3) meters long on the ground. Find the angle of elevation of the sun.
3. An observer 1.5 meters tall is 28.5 meters away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45 degrees. What is the height of the chimney?
4. From a point P on the ground, the angle of elevation of the top of a 10-meter tall building is 30 degrees. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45 degrees. Find the length of the flagstaff.
5. The angle of elevation of the top of a tree from a point on the ground, which is 30 meters away from the foot of the tree, is 30 degrees. Find the height of the tree.