Chapter 3 — Trigonometric Functions

Trigonometric Functions extend trigonometric ratios to all real angles using radian and degree measures, defining six functions (sin, cos, tan, cot, sec, cosec) from unit circle coordinates with applications in physics, engineering, navigation, and seismology.

Chapter 3 covers trigonometric functions, beginning with angle measurement in degrees and radians. The radian measure—angle subtended by arc length 1 in a unit circle—relates to degrees via π radians = 180°. Trigonometric functions are defined using unit circle coordinates: if P(a, b) is on the unit circle at angle x, then cos x = a and sin x = b. Key identities include sin²x + cos²x = 1, and the chapter derives addition/subtraction formulas (cos(x±y), sin(x±y), tan(x±y)), double-angle formulas, triple-angle identities, and product-to-sum conversions. All six functions (sine, cosine, tangent, cotangent, secant, cosecant) have specific domains, ranges, and periodic behaviors analyzed across four quadrants.

1. 01Angle measurement: degree (1/360 revolution) and radian (arc length / radius in unit circle) related by π rad = 180°
2. 02Trigonometric functions defined via unit circle: cos x = x-coordinate, sin x = y-coordinate, with fundamental identity sin²x + cos²x = 1
3. 03Domain and range: sin/cos defined for all reals with range [−1, 1]; tan/cot defined except at odd multiples of π/2 and multiples of π respectively
4. 04Addition formulas: cos(x+y) = cos x cos y − sin x sin y; sin(x+y) = sin x cos y + cos x sin y; tan(x+y) = (tan x + tan y)/(1 − tan x tan y)
5. 05Double-angle formulas: sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x; tan 2x = 2tan x/(1 − tan²x)
6. 06Sum-to-product: cos x + cos y = 2cos((x+y)/2)cos((x−y)/2); sin x + sin y = 2sin((x+y)/2)cos((x−y)/2)