Module 3 · CDS Elementary Mathematics

Trigonometry

Trig ratios, identities, heights & distances, angles.

Trig Ratios · Identities · Height & Distance

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Learning Objectives

Understand the fundamental concepts of Trigonometry
Apply key formulas and techniques to solve problems
Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Trigonometry?

Trigonometry deals with relationships between angles and sides of triangles. In JEE, it covers trigonometric functions, identities, inverse trigonometry, trigonometric equations, and height & distance problems. Weightage is approximately 6-8% in JEE Main.

Key Concepts

1. Trigonometric Functions: sinθ, cosθ, tanθ, cotθ, secθ, cosecθ defined using unit circle. Signs in quadrants: All (I) sin+cos+, Sin (II) sin+others-, Tan (III) tan+cot+, Cos (IV) cos+sec+. Standard angles: 0°,30°,45°,60°,90° values must be memorised.

2. Identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, 1+cot²θ=cosec²θ. Compound angles: sin(A±B)=sinAcosB±cosAsinB, cos(A±B)=cosAcosB∓sinAsinB, tan(A±B)=(tanA±tanB)/(1∓tanAtanB). Double angle: sin2A=2sinAcosA, cos2A=cos²A-sin²A=2cos²A-1=1-2sin²A, tan2A=2tanA/(1-tan²A).

3. Transformation Formulas: 2sinAcosB=sin(A+B)+sin(A-B). 2cosAsinB=sin(A+B)-sin(A-B). 2cosAcosB=cos(A+B)+cos(A-B). 2sinAsinB=cos(A-B)-cos(A+B). Sum→product: sinC+sinD=2sin(C+D)/2 cos(C-D)/2. sinC-sinD=2cos(C+D)/2 sin(C-D)/2.

4. Inverse Trigonometry: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, sec⁻¹x + cosec⁻¹x = π/2. Properties: tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] for xy<1. sin⁻¹(-x) = -sin⁻¹x, cos⁻¹(-x) = π - cos⁻¹x.

5. Trigonometric Equations: General solutions: sinθ=sinα → θ=nπ+(-1)ⁿα. cosθ=cosα → θ=2nπ±α. tanθ=tanα → θ=nπ+α. Always find principal solutions first (0 to 2π).

Solved Examples

Example 1 (Identity — Product to Sum)

Find the value of cos 20° cos 40° cos 80°.

a) 1/4 b) 1/8 c) 1/2 d) 1/16

Solution: b) 1/8. Multiply numerator and denominator by 2 sin 20°: (2 sin 20° cos 20°) cos 40° cos 80°/(2 sin 20°) = (sin 40° cos 40° cos 80°)/(2 sin 20°) = (sin 80° cos 80°)/(4 sin 20°) = sin 160°/(8 sin 20°) = sin(180°-20°)/(8 sin 20°) = sin 20°/(8 sin 20°) = 1/8.

Example 2 (Inverse Trigonometry)

Find the value of tan⁻¹(1/2) + tan⁻¹(1/3).

a) 30° b) 45° c) 60° d) 90°

Solution: b) 45°. tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹[(1/2+1/3)/(1-1/2·1/3)] = tan⁻¹[(5/6)/(5/6)] = tan⁻¹(1) = 45°.

Example 3 (Trigonometric Equation)

Solve: 2sin²θ + sinθ - 1 = 0 for θ in [0, 2π).

a) π/6, 5π/6, 3π/2 b) π/6, 3π/2 c) π/6, 5π/6, π/2 d) π/3, 2π/3, 3π/2

Solution: a) π/6, 5π/6, 3π/2. Let t = sinθ. 2t² + t - 1 = 0 → (2t-1)(t+1) = 0 → t = 1/2 or t = -1. sinθ = 1/2 → θ = π/6, 5π/6. sinθ = -1 → θ = 3π/2. So solutions: π/6, 5π/6, 3π/2.

Shortcut Techniques

For product-to-sum problems, multiply by appropriate sine/cosine terms to create a telescoping chain — as in the cos20°cos40°cos80° example.
Use the 'R formula': a sinθ + b cosθ = R sin(θ+φ) where R = √(a²+b²), φ = tan⁻¹(b/a) — this helps find maximum/minimum values.
Memorise sin18° = (√5-1)/4 and cos36° = (√5+1)/4 — these are frequently tested in JEE.
For inverse trig problems, always check the domain and range of the inverse function before applying formulas.
In trigonometric equations, check for extraneous roots by substituting back into the original equation.

Pro Tip

Trigonometry is the backbone of many JEE calculus problems (integration of trig functions, limits with trig expressions). Master the identities thoroughly — being able to quickly transform expressions using product-to-sum or sum-to-product formulas saves significant time in exams.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Practice Questions

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