CBSE Class 11 Mathematics Chapter 3 Notes: Trigonometric Functions

Chapter Overview

Trigonometric Functions is Chapter 3 of CBSE Class 11 Mathematics and belongs to the Sets and Functions unit (Unit I). It extends the trigonometry you learned in Class 10 to general angles, radian measure, identities, and applications to triangles. This chapter is essential for calculus, coordinate geometry, and physics.

This guide covers every NCERT topic with tables, identities, worked examples, common mistakes, and exam tips.

Angle Measures

Degree and Radian

• Degree: 1 full rotation = 360°. 1° = 60 minutes = 3600 seconds.
• Radian: 1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. 1 full rotation = 2π radians.

Conversion Formulae

Standard Angles Table

Memory tip for sin: sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1 — the pattern is √(0/4), √(1/4), √(2/4), √(3/4), √(4/4). Cosine reverses this pattern.

Signs of Trigonometric Functions in Quadrants (ASTC Rule)

ASTC = "All Silver Tea Cups" or "Add Sugar To Coffee".

Six Trigonometric Functions

Trigonometric Identities

Pythagorean Identities

1. sin²θ + cos²θ = 1 2. 1 + tan²θ = sec²θ 3. 1 + cot²θ = cosec²θ

Sum and Difference Formulas

Double Angle Formulas

Half Angle Formulas

Product-to-Sum and Sum-to-Product

Sine and Cosine Rules

Sine Rule

In any triangle ABC:

a/sin A = b/sin B = c/sin C = 2R

where R is the circumradius of the triangle.

Cosine Rule

In any triangle ABC:

• a² = b² + c² − 2bc cos A
• b² = a² + c² − 2ac cos B
• c² = a² + b² − 2ab cos C

Note: When the angle is 90°, cos 90° = 0 and the cosine rule reduces to the Pythagorean theorem.

Area of a Triangle

• Area = (1/2)ab sin C = (1/2)bc sin A = (1/2)ac sin B
• Heron's formula: Area = √(s(s − a)(s − b)(s − c)) where s = (a + b + c)/2

Important Numericals

1. Proving an Identity

Prove: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan²A + cot²A.

LHS: (sin A + cosec A)² + (cos A + sec A)² = sin²A + 2 sin A cosec A + cosec²A + cos²A + 2 cos A sec A + sec²A = (sin²A + cos²A) + 2(1) + 2(1) + (cosec²A) + (sec²A) = 1 + 2 + 2 + (1 + cot²A) + (1 + tan²A) = 7 + tan²A + cot²A = RHS. Proved.

2. Solving a Trigonometric Equation

Solve: 2 sin²θ + sin θ − 1 = 0 for θ ∈ [0, 2π).

Let x = sin θ. Then 2x² + x − 1 = 0 → (2x − 1)(x + 1) = 0 → x = 1/2 or x = −1 → sin θ = 1/2 → θ = π/6, 5π/6 → sin θ = −1 → θ = 3π/2 Solutions: θ = π/6, 5π/6, 3π/2

3. Sum and Difference Formula

Find the value of sin 75°.

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/(2√2) = (√6 + √2)/4

4. Sine Rule Application

In triangle ABC, a = 10, A = 30°, B = 45°. Find b.

By sine rule: a/sin A = b/sin B → 10/sin 30° = b/sin 45° → 10/(1/2) = b/(1/√2) → 20 = b√2 → b = 20/√2 = 10√2

5. Cosine Rule Application

In triangle ABC, a = 5, b = 7, C = 60°. Find c.

c² = a² + b² − 2ab cos C = 25 + 49 − 2(5)(7)(1/2) = 74 − 35 = 39 c = √39

Conditional Identities

When additional conditions are given on angles, you can derive specific identities.

If A + B + C = π (angles of a triangle):

• sin(A + B) = sin C, cos(A + B) = −cos C
• tan A + tan B + tan C = tan A tan B tan C
• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

If A + B = π/2 (complementary angles):

• sin A = cos B and cos A = sin B
• tan A = cot B and cot A = tan B

Height and Distance Problems

Trigonometric functions solve real-world problems involving heights and distances where direct measurement is impossible.

Procedure

1. Draw a diagram with the object, observer, and line of sight. 2. Identify the angle of elevation (looking up) or angle of depression (looking down). 3. Apply tan θ = opposite/adjacent (or appropriate ratio). 4. Solve for the unknown quantity.

Example: Angle of Elevation

A tower stands vertically on the ground. From a point on the ground 30 m from the foot of the tower, the angle of elevation of the top is 30°. Find the height of the tower.

Let height = h. tan 30° = h/30 → h = 30 tan 30° = 30 × (1/√3) = 30/√3 = 10√3 m ≈ 17.32 m

Example: Angle of Depression

From the top of a 50 m tall building, the angle of depression of a car on the ground is 60°. Find the distance of the car from the building.

Let distance = d. tan 60° = 50/d → d = 50/tan 60° = 50/√3 = 50√3/3 m ≈ 28.87 m

Practice Problems

Problem 1

Prove: (1 + tan A)(1 + tan B) = 2 if A + B = π/4.

Given A + B = π/4, so tan(A + B) = 1. (tan A + tan B)/(1 − tan A tan B) = 1 → tan A + tan B = 1 − tan A tan B → tan A + tan B + tan A tan B = 1 → tan A + tan B(1 + tan A) = 1

Multiply both sides by (1 + tan A): (1 + tan A)(1 + tan B) = 1 + tan A + tan B + tan A tan B = 1 + 1 = 2. Proved.

Problem 2

Solve: cos 2θ + sin θ = 1 for θ ∈ [0, 2π).

cos 2θ = 1 − 2sin²θ, so: 1 − 2sin²θ + sin θ = 1 → −2sin²θ + sin θ = 0 → sin θ(−2sin θ + 1) = 0 → sin θ = 0 → θ = 0, π → sin θ = 1/2 → θ = π/6, 5π/6 Solutions: θ = 0, π/6, 5π/6, π

NCERT Exercise Mapping

Common Mistakes

1. Forgetting radian mode: When using a calculator, always check if it is set to degree or radian mode. An answer of sin 30 = −0.988 means you are in radian mode. 2. Wrong quadrant sign: tan 150° is negative (Quadrant II, only sin/cosec are positive). Many students forget the ASTC rule. 3. Mixing up sum and difference formulas: sin(A − B) = sin A cos B − cos A sin B (minus, not plus). The sign in the argument carries through. 4. Squaring and losing sign information: If sin θ = 1/2, then θ = 30° or 150°. Squaring sin θ = 1/2 gives sin²θ = 1/4 but you lose the quadrant information. 5. Using cosine rule with wrong angle: In c² = a² + b² − 2ab cos C, the angle C must be opposite side c. Mixing up which angle corresponds to which side gives wrong answers. 6. Dividing by zero in tan formulas: tan 90° is undefined because cos 90° = 0. Never write tan 90° = ∞ on the board — write "undefined". 7. Sign errors in double angle formula for cos 2θ: cos 2θ has three forms. Use the one that simplifies your problem: cos²θ − sin²θ, 2cos²θ − 1, or 1 − 2sin²θ.

Exam Tips

• 1-mark questions: Values of trig functions at standard angles, ASTC quadrant signs, or degree-radian conversion.
• 2-mark questions: Prove a simple identity or use a sum/difference formula to find a value.
• 4-mark questions: Prove a compound identity (show every step), solve a trig equation, or apply sine/cosine rule.
• 6-mark questions: Prove a complex identity with multiple steps, or solve a word problem using sine/cosine rule in a triangle.
• Always start identity proofs from the more complex side.
• Write the identity or formula you are using before substituting.
• For trig equations, check that solutions fall within the given interval.
• Draw a rough triangle diagram for sine/cosine rule problems.

Quick Revision Notes

• 1 radian = 180°/π ≈ 57.3°. Arc length l = rθ (θ in radians).
• Standard angles: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2. Cosine reverses.
• ASTC: All positive in Q-I, Sin in Q-II, Tan in Q-III, Cos in Q-IV.
• Pythagorean identities: 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).
• sin 2θ = 2 sin θ cos θ. cos 2θ = cos²θ − sin²θ.
• Sine rule: a/sin A = b/sin B = c/sin C = 2R.
• Cosine rule: a² = b² + c² − 2bc cos A.
• Area = (1/2)ab sin C.