Some Applications of Trigonometry

Main Idea & Core Concepts

Trigonometric ratios are used to calculate the heights of objects and the distances between points without actual measurement, using right-angled triangles.

Key Terms

Line of Sight: The line drawn from the eye of an observer to the point in the object viewed by the observer.
Angle of Elevation: The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.
Angle of Depression: The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.

Step-by-Step Method

Draw a clear diagram representing the given situation.
Mark the given angle(s) of elevation or depression carefully.
Identify the side to be calculated and the known side (opposite, adjacent, or hypotenuse).
Choose the appropriate trigonometric ratio (tan, sin, or cos).
Solve the equation carefully and state the final answer with appropriate units.

Points to Remember

Angle Equality: By alternate interior angles, the angle of depression from an overhead point to an object equals the angle of elevation of the overhead point from the object.

Ratio Selection: Use tan θ when relating height (opposite) and distance (adjacent). Use sin θ or cos θ when the length of a slant surface/line of sight (hypotenuse) is involved.

Important Practice Questions

Find the height of a tower using a given angle of elevation from a point on the ground.
A ladder leans against a wall making an angle with the ground. Find the height reached on the wall using trigonometry.
Solve a two-position observation problem involving angles of elevation or depression to determine the distance between two objects.

Quick Revision

Angle of Elevation

Observer looks upwards. Line of sight is above the horizontal line.

Angle of Depression

Observer looks downwards. Line of sight is below the horizontal line.

Height-Distance Ratio

tan θ = Perpendicular (Height) / Base (Distance)

Key Step

Always convert angles of depression into alternate interior angles at ground level.