Trigonometric functions and circular measure - the Australian ...

{40} • Trigonometric functions and circular measureExampleSketch the graph of y = 3sin4x, for −2π ≤ x ≤ 2π.SolutionThe amplitude of the graph is 3 and the period is 2π 4 = π 2 .The easiest way to draw the graph is to draw one cycle of a sine curve, with amplitude 3,and mark the end point as π 2. We can then extend the graph using periodicity to theinterval −2π ≤ x ≤ 2π.y3y = 3sin 4x2––3 –2 2–7 –5 –34 4 4–4104–1232 23 5 74 4 4x–2–3Cosine graphs of the form y = A cos(nx + α) can be drawn by following the principlesoutlined above for sine graphs.Exercise 17Sketch the graph of y = 2cos3x, for −3π ≤ x ≤ 3π.The graph of tan xThe function tan x has a very different kind of graph to those for the sine and cosinefunctions. The period of tan x is π, rather than 2π, and the amplitude is not defined. Thetangent function is not defined at x = ± π 2, nor at any odd integer multiple of these values.As x approaches π 2from the left, the value of tan x increases without bound. Since tan isan odd function, as x approaches − π 2from the right, tan x decreases without bound.