Board exam focus — Trigonometric Functions (CBSE & HBSE)

The most identity-heavy chapter — every formula reappears in calculus and physics. CBSE allocates 8–10 marks across MCQ, identity proofs, equation solving, and graph sketching. HBSE asks 7–9 marks focused on multiple-angle proofs and trig equations.

Angle Measure & Trig Ratios in All Quadrants

Two Units of Angle Measure

Conversion: π radians = 180° ⇒ 1 rad ≈ 57.296°, 1° = π/180 rad.

Arc Length & Sector Area

For a circle of radius r and central angle θ (in radians):

• Arc length: ℓ = r θ
• Sector area: A = (1/2) r² θ

Signs of Trig Functions in Four Quadrants (ASTC)

All positive | Sin positive | Tan positive | Cos positive (mnemonic: "All Students Take Coffee" / "After School To College").

Standard Angles

Trigonometric Functions of Real Numbers

For any real x, sin x and cos x are defined via the unit circle:

• Domain of sin / cos = ℝ; range = [−1, 1].
• Domain of tan = ℝ − {(2n+1)π/2}; range = ℝ.
• Periods: sin, cos = 2π; tan, cot = π.

Pythagorean Identities

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

CBSE Tip: Always convert between degrees and radians at the start of a problem and stick to one system throughout.

Allied Angles & Identity Reductions

Allied Angles

Angles whose sum/difference with a given angle is a multiple of π/2.

Rule of Thumb

• If the angle is (nπ/2) ± θ with n even, the trig function stays the same.
• If n is odd, the function changes to its co-function (sin ↔ cos, tan ↔ cot, sec ↔ cosec).
• The sign is determined by the quadrant of the resulting angle.

Standard Reductions

Co-function Identities

• sin x = cos(π/2 − x)
• tan x = cot(π/2 − x)
• sec x = cosec(π/2 − x)

Periodicity

• sin(x + 2π) = sin x, cos(x + 2π) = cos x
• tan(x + π) = tan x

Application to Simplification

To evaluate sin(7π/6): note 7π/6 = π + π/6 ⇒ in Q III where sin < 0 ⇒ sin(7π/6) = −sin(π/6) = −1/2.

Useful Identities (Class 11)

• sin²x = (1 − cos 2x)/2
• cos²x = (1 + cos 2x)/2
• 2 sin A cos B = sin(A + B) + sin(A − B)
• 2 cos A sin B = sin(A + B) − sin(A − B)
• 2 cos A cos B = cos(A + B) + cos(A − B)
• 2 sin A sin B = cos(A − B) − cos(A + B)

HBSE Tip: Convert all complex expressions to sin and cos, then simplify — most identities collapse this way.

Sum, Difference & Multiple-Angle Formulas

Sum & Difference Formulas

• 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-Angle Formulas

• sin 2A = 2 sin A cos A = 2 tan A / (1 + tan²A)
• cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1 = (1 − tan²A)/(1 + tan²A)
• tan 2A = 2 tan A / (1 − tan²A)

Triple-Angle Formulas

• sin 3A = 3 sin A − 4 sin³A
• cos 3A = 4 cos³A − 3 cos A
• tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)

Sub-Multiple Angle (A/2)

• sin A = 2 sin(A/2) cos(A/2)
• cos A = 1 − 2 sin²(A/2) = 2 cos²(A/2) − 1
• tan A = 2 tan(A/2) / (1 − tan²(A/2))

Sum-to-Product & Product-to-Sum

Sum to product:

• sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2)
• sin C − sin D = 2 cos((C + D)/2) sin((C − D)/2)
• cos C + cos D = 2 cos((C + D)/2) cos((C − D)/2)
• cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2)

Product to sum: invert the above relations.

Strategy for Identity Proofs

1. Pick the more complicated side; work towards the other.
2. Convert everything to sin / cos.
3. Use Pythagorean and sum-to-product identities to compress.
4. End with the form on the simpler side; write "Hence proved".

CBSE Tip: Memorise the formulas in pairs (sum + difference, double + triple) — boards often ask to verify one and derive the other.

Trigonometric Equations — General Solutions

Periodic Nature ⇒ Infinitely Many Solutions

Trig equations typically have infinitely many solutions; we express them as a general solution using an integer parameter n ∈ ℤ.

Principal Solutions

Solutions lying in [0, 2π) — usually two values (or one when at a quadrant axis).

Standard General Solutions

where n ∈ ℤ and α is the principal solution.

Equations Involving Squares

• sin²x = sin²α ⇒ x = nπ ± α
• cos²x = cos²α ⇒ x = nπ ± α
• tan²x = tan²α ⇒ x = nπ ± α

Strategy

1. Reduce to one of the standard forms above.
2. Find the principal solution α.
3. Write the general solution using the formula.
4. Always state n ∈ ℤ.

Combining Multiple Trig Functions

If the equation mixes sin and cos:

• Use Pythagorean identity to convert.
• Factor when possible.
• Watch out for extraneous solutions introduced by squaring.

Inverse Trig Functions (Preview)

Class 11 only previews them; full treatment is Class 12. For now, treat sin⁻¹, cos⁻¹, tan⁻¹ as functions returning principal angles.

HBSE Tip: After finding the general solution, always substitute n = 0, 1, −1 to list a few specific solutions — examiners reward this verification.

Frequently asked questions

Are these Trigonometric Functions notes free?

Yes — the Trigonometric Functions notes for Mathematics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Trigonometric Functions notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Trigonometric Functions chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Trigonometric Functions.