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📐 Class 12 CBSE: Functions Study Guide

Trigonometric & Special Functions | NCERT Mathematics

Six Trigonometric Functions

sin(x)

Formula: Opposite/Hypotenuse
Period: 2π
Domain: ℝ
Range: [-1, 1]

cos(x)

Formula: Adjacent/Hypotenuse
Period: 2π
Domain: ℝ
Range: [-1, 1]

tan(x)

Formula: Opposite/Adjacent
Period: π
Domain: ℝ - {(2n+1)π/2}
Range: ℝ

cot(x)

Formula: Adjacent/Opposite
Period: π
Domain: ℝ - {nπ}
Range: ℝ

sec(x)

Formula: 1/cos(x)
Period: 2π
Domain: ℝ - {(2n+1)π/2}
Range: (-∞,-1] ∪ [1,∞)

csc(x)

Formula: 1/sin(x)
Period: 2π
Domain: ℝ - {nπ}
Range: (-∞,-1] ∪ [1,∞)

Special Functions

Modulus Function |x|

Definition: |x| = x if x ≥ 0; -x if x < 0
Domain: ℝ
Range: [0, ∞)
Property: V-shaped, always non-negative

Greatest Integer Function ⌊x⌋

Definition: ⌊x⌋ = greatest integer ≤ x
Domain: ℝ
Range: ℤ (integers)
Examples: ⌊2.7⌋ = 2, ⌊-1.5⌋ = -2

Signum Function sgn(x)

Definition: sgn(x) = 1 if x>0; 0 if x=0; -1 if x<0
Domain: ℝ
Range: {-1, 0, 1}
Property: Three-valued piecewise function

📚 Quick Reference Formulas

sin²(x) + cos²(x) = 1

tan(x) = sin(x)/cos(x)

1 + tan²(x) = sec²(x)

1 + cot²(x) = csc²(x)

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) - sin²(x)