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📐 Trigonometric Functions

1. Angles and Measurements

An angle is a measure of rotation of a given ray about its initial point.

π radians = 180° 1 radian = (180/π)°, 1° = (π/180) radians

Length of an arc: l = rθ (where θ is in radians).

2. Trigonometric Functions & Signs

Defined using a unit circle. ASTC Rule (All Sin Tan Cos) helps remember signs in the 4 quadrants:

  • I Quadrant: All are positive.
  • II Quadrant: Sin and Cosec are positive.
  • III Quadrant: Tan and Cot are positive.
  • IV Quadrant: Cos and Sec are positive.

3. Important Trigonometric Identities

Sum and Difference of Angles:

Key Formulas

sin(x ± y) = sin x cos y ± cos x sin y

cos(x ± y) = cos x cos y ∓ sin x sin y

tan(x ± y) = (tan x ± tan y) / (1 ∓ tan x tan y)

Double Angle Formulas:

  • sin 2x = 2 sin x cos x = (2 tan x) / (1 + tan² x)
  • cos 2x = cos² x - sin² x = 2 cos² x - 1 = 1 - 2 sin² x = (1 - tan² x) / (1 + tan² x)
  • tan 2x = (2 tan x) / (1 - tan² x)

4. Transformation Formulas

Sum to Product

sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2)

cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)

cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)