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.​

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 $$\sin(x)=\dfrac12$$​ .

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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 $$180^{\circ}$$. 

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