Schooling · Class 11 · Mathematics · Trigonometric Functions · Chapter on a page
CBSE Class 11 · Mathematics · NCERT Mathematics
Trigonometric Functions: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Radian measure
- Sign convention for angles
- Trigonometric functions as ratios on…
- Domain and range
- Periodicity
- Fundamental identity
- Addition and subtraction formulas
- Formulas and facts
- π rad = 180°
- Arc length s = rθ (θ in radians)
- 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
- Worked examples
- Converting and finding arc length
- Simplifying with identities
- Solving a trigonometric equation
- Watch out for
- Forgetting to convert degrees to…
- Writing sin(A + B) as sin A + sin B
- Giving only the principal solution…
- Using sec θ or tan θ at angles where…
- Mixing up signs in the quadrant rule…
What this chapter is about
This chapter extends your understanding of trigonometry from earlier classes, where you worked mainly with acute angles in right triangles, to a broader framework using any real angle measured in radians. You learn to treat trigonometric ratios as functions defined on the real number line, study their domains, ranges, graphs and periodic behaviour, and solve equations involving them.
Key ideas
- Radian measure: One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full circle is 2π radians, so π radians = 180°.
- Sign convention for angles: Angles measured anticlockwise from the positive x-axis are positive; clockwise angles are negative. This lets every real number correspond to an angle.
- Trigonometric functions as ratios on the unit circle: For a point P(x, y) on the unit circle at angle θ from the positive x-axis, cos θ = x and sin θ = y. The other four functions follow: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ, sec θ = 1 / cos θ, cosec θ = 1 / sin θ.
- Domain and range: sin and cos are defined for all real θ with range [−1, 1]. tan and sec are undefined where cos θ = 0. cot and cosec are undefined where sin θ = 0.
- Periodicity: sin and cos repeat every 2π; tan and cot repeat every π.
- Fundamental identity: sin²θ + cos²θ = 1 for all θ. Dividing through gives 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
- Addition and subtraction formulas: These let you find exact values for sums or differences of angles, for example sin(A + B) = sin A cos B + cos A sin B.
- General solutions of trigonometric equations: Because the functions are periodic, equations have infinitely many solutions expressible in terms of an integer n.
Formulas and facts to remember
- Formula: π rad = 180° · Meaning: Conversion between radians and degrees.
- Formula: Arc length s = rθ (θ in radians) · Meaning: The arc of a circle equals radius times angle in radians.
- Formula: sin²θ + cos²θ = 1 · Meaning: Pythagorean identity; basis for the other two.
- Formula: 1 + tan²θ = sec²θ · Meaning: Derived by dividing the Pythagorean identity by cos²θ.
- Formula: 1 + cot²θ = cosec²θ · Meaning: Derived by dividing the Pythagorean identity by sin²θ.
- Formula: sin(A + B) = sin A cos B + cos A sin B · Meaning: Sine of a sum.
- Formula: cos(A + B) = cos A cos B − sin A sin B · Meaning: Cosine of a sum.
- Formula: tan(A + B) = (tan A + tan B) / (1 − tan A tan B) · Meaning: Tangent of a sum (provided the denominator ≠ 0).
- Formula: sin 2A = 2 sin A cos A · Meaning: Double-angle formula for sine.
- Formula: cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A · Meaning: Three equivalent forms of the double-angle cosine.
- Formula: General solution of sin θ = sin α: θ = nπ + (−1)ⁿ α, n ∈ Z · Meaning: Captures all angles with the same sine value.
- Formula: General solution of cos θ = cos α: θ = 2nπ ± α, n ∈ Z · Meaning: Captures all angles with the same cosine value.
- Formula: General solution of tan θ = tan α: θ = nπ + α, n ∈ Z · Meaning: Captures all angles with the same tangent value.
Common mistakes
- Forgetting to convert degrees to radians before using the arc-length formula s = rθ → always check that θ is in radians when the formula requires it.
- Writing sin(A + B) as sin A + sin B → remember the addition formula involves products sin A cos B and cos A sin B.
- Giving only the principal solution of a trigonometric equation and ignoring the general solution → state the family of solutions using n ∈ Z.
- Using sec θ or tan θ at angles where cos θ = 0 → first check the domain; these functions are undefined at θ = π/2 + nπ.
- Mixing up signs in the quadrant rule (ASTC) → draw a quick unit-circle sketch to confirm which function is positive or negative in each quadrant.
Quick revision
- Radian measure: π rad = 180°; arc length s = rθ with θ in radians.
- Pythagorean identity sin²θ + cos²θ = 1 is the foundation of all trigonometric identities.
- sin and cos have period 2π and range [−1, 1]; tan has period π and range all real numbers.
- Use the ASTC rule (All, Sin, Tan, Cos positive in quadrants I–IV) to fix signs.
- General solutions involve an integer n; memorise the standard forms for sin θ = sin α, cos θ = cos α, tan θ = tan α.