Trig graphs: $\sin x$ , $\cos x$ , $\tan x$

In a nutshell

The graphs of $y=\sin(x)$ , $y=\cos(x)$ and $y=\tan(x)$ are periodic graphs - which means they repeat after a regular interval. By learning these intervals, it is possible to sketch the graphs between a range of given angles and deduce the values of $\sin(x)$ , $\cos(x)$ and $\tan(x)$ for certain values of $x$ .

Graphs of $\sin x$ and $\cos x$

The graphs of $y=\sin(x)$ and $y=\cos(x)$ are very similar. They repeat every $360^{\circ}$ with a maximum of $1$ and minimum of $-1$ .

The graph of $y=\sin(x)$ crosses the $x$ -axis at $-180^{\circ}$ , $0^{\circ}$ , $180^{\circ}$ , $360^{\circ}$ and so on.

The graph of $y=\cos(x)$ crosses the $x$ -axis at $-90^{\circ}$ , $90^{\circ}$ , $270^{\circ}$ , $450^{\circ}$ and so on.

Maths; Trigonometry; KS5 Year 12; Trig graphs: sin x, cos x, tan x

Maths; Trigonometry; KS5 Year 12; Trig graphs: sin x, cos x, tan x

Note: The graph of $y=\sin(x)$ translated $90^{\circ}$ to the left is the same as $y=\cos(x)$ .

Example 1

Given that $\sin(30^{\circ}) = \dfrac12$ , use the graph of $y=\sin(x)$ to other values of $x$ for which $\sin(x)=\dfrac12$ .

Maths; Trigonometry; KS5 Year 12; Trig graphs: sin x, cos x, tan x

Sketch the horizontal line $y=\dfrac{1}{2}$ and use symmetry to find other intersection points.

The first point will be $30^{\circ}$ to the left of $180^{\circ}$ .

$180-30 = 150^{\circ}$

Because $\sin(x)$ repeats every $360^\circ$ , every point $360^{\circ}$ to the left and right of $30^{\circ}$ and $150^{\circ}$ will also be a solution.

$\underline{x=-330^{\circ},-210^{\circ},150^\circ,390^{\circ},510^{\circ}....}$

Graphs of $\tan x$

The graph of $y=\tan(x)$ approaches $-\infin$ and $+\infin$ at the asymptotes at $-90^{\circ}$ and $90^{\circ}$ respectively. The graph crosses the $x$ -axis at the origin. The graph is periodic - repeating every $180^{\circ}$ .

Maths; Trigonometry; KS5 Year 12; Trig graphs: sin x, cos x, tan x