Questions
3 questions
Difficulty
Medium
Importance
High yield for SSC CGL/CHSL
Overview
Trigonometry Heights and Distances involves the application of trigonometric ratios to solve real-world measurement problems without physical access to the objects. Mastering this is vital for SSC and banking exams as it tests both conceptual geometric visualization and rapid calculation skills under time pressure.
Core Trigonometric Ratios
These ratios form the foundation of all heights and distances problems by relating the angles of a right-angled triangle to the lengths of its sides. In exams, you should memorize these ratios relative to the angle theta to avoid drawing triangles from scratch.
- sin(theta) = Perpendicular / Hypotenuse
- cos(theta) = Base / Hypotenuse
- tan(theta) = Perpendicular / Base
- cosec(theta) = 1/sin(theta)
- sec(theta) = 1/cos(theta)
- cot(theta) = 1/tan(theta)
Angle of Elevation and Depression
The angle of elevation is measured upwards from the horizontal, while the angle of depression is measured downwards. Crucially, the angle of depression from a point is always equal to the angle of elevation from the target point due to alternate interior angles.
- Elevation: Angle between horizontal line of sight and object above
- Depression: Angle between horizontal line of sight and object below
- Key Geometric Property: Angle of depression = Angle of elevation
- Horizontal line remains parallel in both cases
Standard Triangle Ratios (Speed Solving)
Competitive exams favor specific angles like 30, 45, and 60 degrees. Using the fixed ratio method is significantly faster than calculating the trigonometric functions manually during an exam.
- 30-60-90 Triangle: Sides are 1 : root(3) : 2
- 45-45-90 Triangle: Sides are 1 : 1 : root(2)
- tan(30) = 1/root(3)
- tan(45) = 1
- tan(60) = root(3)
Formula Sheet
tan(theta) = height / distance
h = d * tan(theta)
d = h * cot(theta)
sin^2(theta) + cos^2(theta) = 1
1 + tan^2(theta) = sec^2(theta)
1 + cot^2(theta) = cosec^2(theta)
Exam Tip
Memorize the 1:root(3):2 and 1:1:root(2) ratios to skip writing out sin/cos/tan equations entirely during the exam.
Common Mistakes
- Confusing the base and perpendicular when the angle is given with the vertical instead of the horizontal.
- Neglecting the height of the observer when the problem explicitly states it.
- Using sin or cos instead of tan, which requires an extra step involving the hypotenuse.
More Revision Notes
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