Trigonometric Functions: Definition, Domain, Range and Quadrant Signs

In earlier classes, sine, cosine, and tangent were defined only for acute angles using the sides of a right triangle. CBSE Class 11 Maths Chapter 3 extends this idea to any angle whatsoever, using the unit circle. This generalisation is what turns “trigonometric ratios” into “trigonometric functions,” and it is the foundation for everything from graphing sine waves to solving trigonometric equations in Class 12.

Defining Trigonometric Functions Using the Unit Circle

Consider a unit circle (radius 1) centred at the origin of the coordinate axes. Let P(a, b) be any point on the circle such that the angle AOP = x radians, meaning the arc AP has length x.

We define:

cos x = a and sin x = b

Since triangle OMP is right-angled, OM² + MP² = OP², which gives a² + b² = 1. This directly proves the most important identity in trigonometry:

cos²x + sin²x = 1, for every real number x

Because a and b are simply coordinates of a point on the unit circle, sin x and cos x are defined for every real value of x, positive, negative, or zero — not just acute angles.

Values at Quadrantal Angles

Angles that are integral multiples of π/2 are called quadrantal angles. Since one complete revolution subtends 2π radians, the points A(1,0), B(0,1), C(−1,0), and D(0,−1) correspond to angles 0, π/2, π, and 3π/2. This gives:

• cos 0° = 1, sin 0° = 0

• cos(π/2) = 0, sin(π/2) = 1

• cos π = −1, sin π = 0

• cos(3π/2) = 0, sin(3π/2) = −1

• cos 2π = 1, sin 2π = 0

An important consequence: sin x = 0 whenever x = nπ, and cos x = 0 whenever x = (2n+1)π/2, where n is any integer.

The Other Four Trigonometric Functions

Once sine and cosine are defined, the remaining four functions follow as ratios:

• cosec x = 1/sin x, x ≠ nπ

• sec x = 1/cos x, x ≠ (2n+1)π/2

• tan x = sin x / cos x, x ≠ (2n+1)π/2

• cot x = cos x / sin x, x ≠ nπ

From cos²x + sin²x = 1, dividing through by cos²x and sin²x respectively gives two more identities you will use constantly:

1 + tan²x = sec²x and 1 + cot²x = cosec²x

Sign of Trigonometric Functions in Each Quadrant

Since cos x = a and sin x = b are just coordinates of a point on the unit circle, their signs follow the sign of the coordinates in each quadrant. Using ∠AOQ = −x to show cos(−x) = cos x and sin(−x) = −sin x, we get the complete ASTC-style sign table:

A handy way to remember this is the phrase “All Sin Tan Cos” (ASTC), read quadrant by quadrant — in quadrant I, All functions are positive; in II, only Sin (and cosec); in III, only Tan (and cot); in IV, only Cos (and sec).

Domain and Range of Each Function

• y = sin x and y = cos x: domain is all real numbers; range is the interval [−1, 1].

• y = cosec x: domain excludes x = nπ; range is y ≤ −1 or y ≥ 1.

• y = sec x: domain excludes x = (2n+1)π/2; range is y ≤ −1 or y ≥ 1.

• y = tan x: domain excludes x = (2n+1)π/2; range is all real numbers.

• y = cot x: domain excludes x = nπ; range is all real numbers.

Notice that sine and cosine are the only two functions with a bounded range — this is why they are the natural choice for modelling periodic phenomena like tides or sound waves, which never exceed a fixed amplitude.

Periodicity: Why Values Repeat After 2π (or π)

Since one full revolution returns point P to the same position, sin(2nπ + x) = sin x and cos(2nπ + x) = cos x for any integer n. Because tan x is a ratio of sin and cos, its values repeat over a shorter interval — every π radians, not 2π. This is exactly why tan x has vertical asymptotes wherever cos x = 0, and it is essential for later chapters on solving general trigonometric equations.

Master the Unit Circle with Guided Practice

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Test Your Knowledge

Q1.In the third quadrant (π < x < 3π/2), which trigonometric functions are positive?

sin x and csc x onlytan x and cot x onlycos x and sec x onlyAll trigonometric functions

Q2.The domain of the tangent function f(x) = tan x is:

R − {nπ : n ∈ Z}[−1, 1]R − {(2n + 1)π / 2 : n ∈ Z}R (all real numbers)

Q3.What is the range of the sine function f(x) = sin x for all x ∈ R?

(−∞, ∞)[0, 1][−π, π][−1, 1]