Questions
5–8 questions in board exams
Difficulty
Medium
Importance
Core — never skip
Overview
Trigonometric functions form the backbone of calculus and wave theory in the CBSE Class 11-12 curriculum. Mastering this topic is essential because it bridges the gap between geometry and algebraic analysis, appearing frequently in both competitive entrance exams and board papers.
Trigonometric Identities
Identities serve as the foundation for simplifying complex trigonometric expressions. Understanding the fundamental Pythagorean relations and sum-to-product conversions is vital for solving integral and derivative problems in later chapters.
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
General Solutions
General solutions provide the set of all possible values for a variable that satisfies a trigonometric equation. Exams often test the ability to derive specific solutions within a given interval after determining the general expression.
- sin θ = 0 implies θ = nπ
- cos θ = 0 implies θ = (2n + 1)π / 2
- sin θ = sin α implies θ = nπ + (-1)ⁿα
- cos θ = cos α implies θ = 2nπ ± α
- tan θ = tan α implies θ = nπ + α
Properties of Triangles
This section utilizes the sine rule, cosine rule, and projection formulas to solve for unknown sides or angles of a triangle. These identities are frequently applied in geometry-heavy problems and vector analysis.
- Sine Rule: a/sin A = b/sin B = c/sin C = 2R
- Cosine Rule: a² = b² + c² - 2bc cos A
- Projection Rule: a = b cos C + c cos B
- Napier's Analogy: tan((B-C)/2) = ((b-c)/(b+c)) cot(A/2)
- Area of Triangle = 1/2 bc sin A
Formula Sheet
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ
tan(2θ) = 2 tan θ / (1 - tan²θ)
sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)
cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)
2 sin A cos B = sin(A+B) + sin(A-B)
2 cos A cos B = cos(A+B) + cos(A-B)
2 sin A sin B = cos(A-B) - cos(A+B)
Exam Tip
Always convert all functions to sine and cosine when stuck; most complex identities resolve quickly once expressed in terms of basic ratios.
Common Mistakes
- Forgetting to include the integer constant 'n' when writing general solutions for trigonometric equations.
- Misapplying the sign convention in quadrants, especially regarding the periodicity of tangent and cotangent functions.
- Neglecting the domain restrictions of inverse trigonometric identities during simplification steps.
More Revision Notes
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