Section 4.6 Graphs of Other Trigonometric Functions MA.PC.4.5 2000 Define and graph trigonometric functions (i.e.,sine, cosine, tangent, cosecant, secant, cotangent). MA.PC.4.6 2000 Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. MA.PC.4.7 2000 Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift.

Graph of y=tan x

The Tangent Curve Period: π Notice that the graph passes through -1 a quarter of the way through the period and 1 at three quarters of the way. Period: π Domain: All real numbers except odd multiples of π/2 Range: All real numbers Vertical asymptotes: at odd multiples of π/2 Point out that it goes through (1,pi/4) and (-1, -pi/4)

Graphing y = A tan(Bx + C) + D Vertical Shift = D: Moves the midline Amplitude = |A|: Tangent crosses the amplitude (max and min lines) at ¼ and3/4 of the way between the asymptotes Period = /B: An asymptote occurs at /2B Phase Shift = -C/B: Shifts graph right or left

Example Graphs

Example

Example

The Cotangent Curve Period: π Notice the graph passes through 1 a quarter of the way through the period and -1 at three quarters of the way. Period: π Domain: All real #s except integral multiples of π Range: All real numbers Vertical asymptotes: at integral multiples of π

Notice the relationship to sin x The Cosecant Curve Notice the relationship to sin x Period: 2π Domain: All real #s except integral multiples of π Range: (-∞,-1]U[1,∞] Vertical asymptotes: at integral multiples of π

Notice the relationship to cos x The Secant Curve Notice the relationship to cos x Period: 2π Domain: All real numbers except odd multiples of π/2 Range: (-∞,-1]U[1,∞] Vertical asymptotes: at odd multiples of - π/2

Example

Example

The Six Curves of Trigonometry

Graphs of the six trig functions

Graphs of the six trig functions

Review (a) (b) (c) (d)

Review (a) (b) (c) (d)

Homework Page 546-547 1-6, 13-16, 25, 28, 42, 44