Graphs of trig functions

See also transformations of functions.

Contents

Trig functions to graph

Using unit circle, sin⁡\sin, cos⁡\cos and tan⁡\tan can be extended to all real numbers (with some asymptotes for tan⁡\tan) in a way that they are periodic, and that sin⁡\sin and tan⁡\tan are odd while cos⁡\cos is even.

y=sin⁡xy = \sin x passes through (0,0), (π2,1), (π,0), (3π2,−1), (2π,0)(0, 0),\, (\frac{\pi}{2}, 1),\, (\pi, 0),\, (\frac{3\pi}{2}, -1),\, (2\pi, 0).

y=cos⁡xy = \cos x passes through (0,1), (π2,0), (π,−1), (3π2,0), (2π,1)(0, 1),\, (\frac{\pi}{2}, 0),\, (\pi, -1),\, (\frac{3\pi}{2}, 0),\, (2\pi, 1).

Both sin⁡x\sin x and cos⁡x\cos x have a period of 2π2\pi, and the pattern repeats for other values of xx. These are the five points you should consider when graphing or sketching their transformations.

y=tan⁡xy = \tan x are not usually used for transformations, but just remember the graph looks like lots of //, with vertical asymptotes at every odd multiples of π2\frac\pi2.

Example: Given

f(x)=−3cos⁡(2(x+π3))+1f(x) = -3\cos\left(2\left(x + \frac{\pi}{3}\right)\right) + 1

Find the coordinates of a maximum and a minimum.

Compare to base function x↦cos⁡xx \mapsto \cos x, the transformations are

  1. horizontal stretch by 12\frac12
  2. vertical reflection across xx-axis and vertical stretch by a factor of 33.
  3. horizontal translation by π3\frac{\pi}{3} to the left.
  4. vertical translation by 11 up.

The base function has a maximum at (0,1)(0, 1) and minimum at (π,−1)(\pi, -1).

Upon a reflection, a maximum become a minimum, vice versa.

The corresponding minimum is

(12⋅0−π3,−3(1)+1)=(−π3,−2)■{\left(\frac{1}{2}\cdot 0 - \frac{\pi}{3}, -3(1) + 1\right) = \left( - \frac{\pi}{3}, -2\right)} \qed

The corresponding maximum is

(12⋅π−π3,−3(−1)+1)=(π6,4)■{\left(\frac{1}{2}\cdot \pi - \frac{\pi}{3}, -3(-1) + 1\right) = \left( \frac{\pi}{6}, 4\right)} \qed

From graph or description to function

For sinusoidal functions in the forms

f(x)=Asin⁡(ω(x−h))+kf(x) = A\sin(\omega(x - h)) + k
g(x)=Acos⁡(ω(x−h))+kg(x) = A\cos(\omega(x - h)) + k
symbol formula notes
AA max−min2\displaystyle \frac{\text{max} - \text{min}}{2} amplitude: distance between average and each max or min
kk max+min2\displaystyle\frac{\text{max} + \text{min}}{2} vertical shift: average: y=ky = k is the line about which the function is oscillating.
hh horizontal shift: depends on a given point in the question
ω\omega 1horiz. stretch factor\displaystyle \frac{1}{\text{horiz. stretch factor}} angular frequency: the number of cycles in 2π2\pi
TT 2πω\displaystyle \frac{2\pi}{\omega} period: the difference in xx between consecutive complete cycles.

In modelling, time tt is often used instead of xx.

HL: Inverse trig functions

See also domain and range of inverse functions and domain restrictions

HL students should also know the graphs of arcsin⁡,arccos⁡,arctan⁡\arcsin, \arccos, \arctan, commonly known as sin⁡−1,cos⁡−1,tan⁡−1\sin^{-1}, \cos^{-1}, \tan^{-1}.

Note that all 3 are invertible functions, while sin⁡\sin, cos⁡\cos, and tan⁡\tan are not.

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