Questions
2–4 questions per board paper
Difficulty
Medium
Importance
High yield for board exams
Overview
Some Applications of Trigonometry focuses on calculating heights and distances of inaccessible objects using trigonometric ratios. Mastery of this topic is essential for translating word problems into accurate right-angled triangle diagrams. It serves as a fundamental building block for higher engineering mathematics like surveying and navigation.
Line of Sight and Angles
The line of sight is the imaginary line drawn from the observer's eye to the point on the object viewed. Understanding the orientation relative to the horizontal level is the first step in constructing a valid triangle for any problem.
- Angle of elevation is measured upward from the horizontal
- Angle of depression is measured downward from the horizontal
- The angle of elevation equals the angle of depression due to alternate interior angles
- Always define the horizontal line parallel to the ground
Solving Right-Angled Triangles
Most problems require solving a right-angled triangle where one side and one acute angle are known. You must effectively select the correct trigonometric ratio based on the relationship between the angle and the sides involved.
- SOH-CAH-TOA rule for ratio identification
- tan(θ) = Opposite / Adjacent (most common ratio)
- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- Use exact values for standard angles: 30, 45, and 60 degrees
Advanced Two-Triangle Problems
More complex exam questions involve two separate angles of elevation or depression from two different points. These problems require setting up a system of linear equations to solve for the common height.
- Create two separate triangles sharing a common side (usually the height)
- Express the horizontal base as a sum or difference of segments
- Use substitution to eliminate the unknown horizontal distance
- Solve for the height 'h' using the intersection of two tangent ratios
Formula Sheet
tan(θ) = Height / Base
sin(θ) = Height / Hypotenuse
cos(θ) = Base / Hypotenuse
cot(θ) = 1 / tan(θ)
Exam Tip
Always sketch a clean diagram, mark the known angle and side, and label the height as 'h' before attempting any algebraic calculations.
Common Mistakes
- Interchanging the angle of elevation and angle of depression in the diagram
- Failing to account for the height of the observer when the observer is not at ground level
- Using the wrong trigonometric ratio (e.g., using sin instead of tan) when the hypotenuse is unknown
More Revision Notes
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