Graphs of Basic Trigonometric Functions (SQA National 5 Maths): Revision Note

Graphs of the three basic trig functions

What trig graphs should I be familiar with?

• You should be familiar with the graphs of the following trigonometric functions

  • $y=\sin x$

  • $y=\cos x$

  • $y=\tan x$

• The variable $x$ is like an angle

  • but the angle can go beyond acute to become obtuse or reflex

    • $0°\le x\le 360°$

  • $x$ can even be negative, or take on values greater than 360°

• Trig graphs have repeating (periodic) shapes and symmetries that you need to know

What does the graph of y = sin x look like?

• The graph of $y=\sin x$ is a wave that

  • oscillates between heights of 1 and -1

    • It has an amplitude of 1 (i.e. it goes a maximum of 1 above and below its 'middle line' on the $x$-axis)

  • and repeats every 360° (its period is 360°)

• It goes through the origin, (0, 0)

  • Then every 90° it cycles through the heights 1, 0, -1, 0, ...

What does the graph of y = cos x look like?

• The graph of $y=\cos x$ is a wave that

  • oscillates between heights of 1 and -1

    • It has an amplitude of 1 (i.e. it goes a maximum of 1 above and below its 'middle line' on the $x$-axis)

  • and repeats every 360° (its period is 360°)

• It has a y-intercept of 1, coordinates (0, 1)

  • Then every 90° it cycles through the heights 0, -1, 0, 1, ...

• $y=\cos x$ is the same as translating $y=\sin x$ by 90 to the left

What does the graph of y = tan x look like?

• The graph of $y=\tan x$ is not a wave but consists of branches

  • that repeat every 180° (its period is 180°)

    • This is half the period of $\sin x$ and $\cos x$

• $y=\tan x$ is sometimes drawn with dotted vertical lines that separate the branches

  • These are called asymptotes

    • These occur every 180° at $x=90°$, $x=270°$, ... (and $x=-90°$, $x=-270°$, ...)

    • The graph does not touch these lines, but gets closer and closer to them

• $y=\tan x$ has no maximum or minimum values

  • It can take on any positive or negative value (as well as zero)

  • A branch heads down towards $-\infty$ on the right side of the asymptotes, and heads up towards $+\infty$ on the left side of the asymptotes

• $y=\tan x$ goes through the origin, (0, 0)