Trigonometric Functions – Class 11 Maths (NCERT Based)
The chapter Trigonometric Functions is a core topic in Class 11 Mathematics. It introduces students to the concept of angles and trigonometric ratios, which are used widely in higher mathematics, physics, and engineering. This chapter forms a strong base for calculus and coordinate geometry.
📌 Measurement of Angles
Angles can be measured in:
- Degree Measure
- Radian Measure
The relation between them is:
π radians = 180°
Radian measure is more commonly used in advanced mathematics.
✍️ Trigonometric Ratios
For an angle θ in a right-angled triangle:
- 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 Identities
Important identities from NCERT include:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
These identities help simplify trigonometric expressions.
🔁 Trigonometric Functions of Angles
NCERT covers:
- Trigonometric functions of sum and difference of angles
- Trigonometric functions of multiple angles
- Values of trigonometric functions in different quadrants
ASTC Rule (Easy to Remember)
| Quadrant | Name | Positive Trigonometric Ratios |
|---|---|---|
| I (0°–90°) | A – All | All ratios are positive |
| II (90°–180°) | S – Sine | sin θ, cosec θ |
| III (180°–270°) | T – Tangent | tan θ, cot θ |
| IV (270°–360°) | C – Cosine | cos θ, sec θ |
ASTC = All Students Take Coffee ☕
rigonometric Functions – Important Rules (Class 11 Maths)
🔹 1. Reciprocal Rules
These rules show the relationship between trigonometric ratios:
- cosec θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
🔹 2. Quotient Rules
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
🔹 3. Pythagorean Identity Rules
These identities are very important and used frequently:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
🔹 4. Co-function Rules
For complementary angles (90° − θ):
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
- cosec(90° − θ) = sec θ
- sec(90° − θ) = cosec θ
🔹 5. Negative Angle Rules
- sin(−θ) = −sin θ
- cos(−θ) = cos θ
- tan(−θ) = −tan θ
- cosec(−θ) = −cosec θ
- sec(−θ) = sec θ
🔹 6. Periodicity Rules
- sin(θ + 2π) = sin θ
- cos(θ + 2π) = cos θ
- tan(θ + π) = tan θ
🔹 7. Sum and Difference Rules
- 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)
🔹 8. Double Angle Rules
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ
- tan 2θ = 2 tan θ / (1 − tan²θ)
🔹 9. Quadrant Sign Rules (ASTC Rule)
- 1st Quadrant: All ratios are positive
- 2nd Quadrant: sin θ is positive
- 3rd Quadrant: tan θ is positive
- 4th Quadrant: cos θ is positive
(ASTC = All, Sine, Tangent, Cosine)
🔹 10. Standard Angle Values Rule
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Not defined |