Mathematics · Class 11
Trigonometric Functions
Learn Trigonometric Functions with notes, examples, and practice questions.
4 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Trigonometry extends beyond right triangles to circular functions defined on real numbers. Radian measure, identities, and graphs of sin, cos, tan are central to Class 11 and 12.
Radians & Arc Length
Conversion
π rad = 180°
Arc length
l = rθ (θ in radians)
Key Identities
Pythagorean
sin²x + cos²x = 1
Addition
sin(A±B), cos(A±B)
Double angle
sin 2x = 2 sin x cos x
General Solutions
- sin x = sin α → x = nπ + (−1)ⁿα
- cos x = cos α → x = 2nπ ± α
- tan x = tan α → x = nπ + α
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
sin 75°
Find exact value.
- 175°=45°+30°
- 2Use sin(A+B)
Answer
(√6+√2)/4
General solution
sin x = 1/2.
- 1α=π/6
- 2x = nπ + (−1)ⁿ(π/6)
Answer
nπ + (−1)ⁿπ/6
Key Points to Remember
- ✓ Memorise ASTC sign rule
- ✓ Principal and general solution distinction
- ✓ Graphs: period 2π for sin/cos
Exam Tips
- • Convert degrees to radians in calculus chapters
- • Use identities before expanding
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Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Reciprocal & Quotient
- sin θ = 1/cosec θ ; cosec θ = 1/sin θ
- cos θ = 1/sec θ ; sec θ = 1/cos θ
- tan θ = 1/cot θ ; cot θ = 1/tan θ
- tan θ = sin θ/cos θ ; cot θ = cos θ/sin θ
Pythagorean Identities
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ
Angle Addition
- 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)
Double & Triple Angle
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
- tan 2θ = 2 tan θ/(1 − tan²θ)
- sin 3θ = 3 sin θ − 4 sin³θ
- cos 3θ = 4 cos³θ − 3 cos θ
Power Reducing (Integrals)
- sin²θ = (1 − cos 2θ)/2
- cos²θ = (1 + cos 2θ)/2
- tan²θ = (1 − cos 2θ)/(1 + cos 2θ)
Product ↔ Sum
- sin A sin B = ½[cos(A−B) − cos(A+B)]
- cos A cos B = ½[cos(A−B) + cos(A+B)]
- sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
- cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
Half Angle & Triangle Laws
- sin(θ/2) = ±√[(1 − cos θ)/2]
- Arc length: S = rθ (θ in radians)
- a/sin A = b/sin B = c/sin C (sine rule)
- a² = b² + c² − 2bc cos A (cosine rule)
Practice MCQs — Trigonometric Functions
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
π radians equals: