Trigonometric Functions Class 11 Math Quick Recall / Short Notes (2026-27)
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[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)
- Angle Conversion: π radians = 180°. (Radian = Degree × π/180).
- Arc Length: l = rθ (where θ is in radians).
- ASTC Rule:
- All (+ve) in 1st Quad.
- Sine (+ve) in 2nd Quad.
- Tan (+ve) in 3rd Quad.
- Cos (+ve) in 4th Quad.
- Identities: sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = cosec²x.
- Periodicity: Sin, Cos, Sec, Cosec repeat after 2π. Tan and Cot repeat after π. 📥 Download 1-Page Short Notes PDF (Zero-Friction)
Introduction
Trigonometric Functions extend the geometry of right triangles to the circular motion of the Unit Circle, defining the periodic nature of waves and oscillations. Master radian measure, ASTC rules, and the "Core 10" identities to solve complex rotation problems in physics and engineering. This Class 11 Math Chapter 3 guide ensures you have all essential formulas for JEE and Board success. Trigonometry is the study of triangles, but in Chapter 3, it evolves into the study of periodic motion.
1. Measuring Angles: Degrees vs Radians
- Degree Measure: If a rotation is 1/360th of a revolution, the angle is 1°.
- Radian Measure: Angle subtended at the center by an arc of length 1 unit in a unit circle.
- Relation: 2π radians = 360°.
- 1 Radian ≈ 57° 16'.
- Angle in Radians = (π/180) × Degree.
2. The Unit Circle and ASTC Rule
The definitions of trigonometric functions are extended to all real numbers using a unit circle.
- Signs of Functions:
- Quadrant I: All (Sin, Cos, Tan, Cot, Sec, Cosec) are Positive.
- Quadrant II: Sine and Cosecant are Positive.
- Quadrant III: Tangent and Cotangent are Positive.
- Quadrant IV: Cosine and Secant are Positive.
3. Domain and Range of Trig Functions
| Function | Domain | Range |
|---|---|---|
| sin x | R | [-1, 1] |
| cos x | R | [-1, 1] |
| tan x | R - {(2n+1)π/2} | R |
| cot x | R - {nπ} | R |
| sec x | R - {(2n+1)π/2} | (-∞, -1] ∪ [1, ∞) |
| cosec x | R - {nπ} | (-∞, -1] ∪ [1, ∞) |
4. Essential Trigonometric Formulas
Sum and Difference of Angles:
- sin(x ± y) = sin x cos y ± cos x sin y
- cos(x ± y) = cos x cos y ∓ sin x sin y
- tan(x ± y) = (tan x ± tan y) / (1 ∓ tan x tan y)
Double Angle Formulas:
- sin 2x = 2 sin x cos x = 2 tan x / (1 + tan²x)
- cos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x = (1 - tan²x) / (1 + tan²x)
- tan 2x = 2 tan x / (1 - tan²x)
5. Trigonometric Equations
- sin x = 0 => x = nπ, where n ∈ Z.
- cos x = 0 => x = (2n + 1)π/2.
- General Solutions:
- sin x = sin y => x = nπ + (-1)ⁿ y.
- cos x = cos y => x = 2nπ ± y.
- tan x = tan y => x = nπ + y.
Comprehensive Exam Strategy (Q&A)
Q1: Find the value of sin(765°). Answer:
- sin(765°) = sin(2 × 360° + 45°)
- Since sin(n × 360° + x) = sin x,
- sin(765°) = sin(45°) = 1/√2.
Q2: Find the radian measure of -47° 30'. Answer:
- 30' = (1/2)° = 0.5°.
- So, -47° 30' = -47.5°.
- In radians = -47.5 × (π/180) = -19π/72.
Q3: Prove that cos³x + sin³x / cos x + sin x = 1 - sin x cos x. Answer:
- Using a³ + b³ = (a + b)(a² - ab + b²):
- LHS = (cos x + sin x)(cos²x - sin x cos x + sin²x) / (cos x + sin x)
- = cos²x + sin²x - sin x cos x
- = 1 - sin x cos x. Proved.
Related Revision Notes
- Chapter 2: Relations and Functions
- Chapter 5: Complex Numbers
- [External Reference: NCERT Class 11 Math Chapter 3 (Authoritative Source)]
Conclusion
Trigonometry is the bridge between geometry and algebra. By mastery the "Core 10" formulas and visualizing the Unit Circle, you unlock the ability to solve complex rotation and wave problems. Keep your radians straight and your ASTC rule handy!