Questions
5–8 questions in board exams
Difficulty
Medium
Importance
Core — never skip
Overview
Introduction to Trigonometry establishes the mathematical relationship between the sides and angles of right-angled triangles. It is a cornerstone of higher mathematics, frequently tested in board exams and competitive engineering entrances through application-based problems and identity proofs.
Trigonometric Ratios
Trigonometric ratios define the proportions between sides of a right triangle with respect to an acute angle. Mastering these ratios is essential for solving problems involving heights, distances, and geometric constructions.
- sin(θ) = Perpendicular / Hypotenuse
- cos(θ) = Base / Hypotenuse
- tan(θ) = Perpendicular / Base
- cosec(θ) = 1 / sin(θ)
- sec(θ) = 1 / cos(θ)
- cot(θ) = 1 / tan(θ)
Trigonometric Values for Specific Angles
Memorizing standard values for specific angles is critical for rapid calculation in objective-type questions. These values recur frequently in geometric and physics-based numericals.
- sin(0, 30, 45, 60, 90) = 0, 1/2, 1/√2, √3/2, 1
- cos(0, 30, 45, 60, 90) = 1, √3/2, 1/√2, 1/2, 0
- tan(0, 30, 45, 60, 90) = 0, 1/√3, 1, √3, Not Defined
Trigonometric Identities
These fundamental equalities relate squares of ratios and are the most common tools for simplifying complex trigonometric expressions and proving geometric theorems.
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = cosec²(θ)
- tan(θ) = sin(θ) / cos(θ)
- cot(θ) = cos(θ) / sin(θ)
Formula Sheet
sin(θ) = P/H
cos(θ) = B/H
tan(θ) = P/B
sin²(θ) + cos²(θ) = 1
sec²(θ) - tan²(θ) = 1
cosec²(θ) - cot²(θ) = 1
tan(θ) = sin(θ)/cos(θ)
cot(θ) = cos(θ)/sin(θ)
Exam Tip
Always draw a rough sketch of the right-angled triangle when solving word problems to visualize the angle of elevation or depression accurately.
Common Mistakes
- Confusing reciprocal identities like thinking sec(θ) is 1/sin(θ) instead of 1/cos(θ).
- Neglecting the domain restrictions, specifically division by zero when the denominator is zero in tan(90) or cot(0).
- Incorrectly squaring the argument instead of the ratio, e.g., writing sin²(θ²) instead of (sin θ)².
More Revision Notes
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