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Trigonometric graphs - Higher

Trigonometric graphs - Higher

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Summary

Trigonometric graphs

​​In a nutshell

Trigonometric graphs are a class of "wave" graphs. They repeat themselves after a point, which makes them periodic. There are three main trigonometric graphs: sine, cosine and tangent and there are key points on each to learn.



Sine graph


The graph of y=sin(x)y=\sin(x) is pictured above. The xx-axis displays degrees, the unit of angle. You can see that the graph passes through the origin and has peaks where y=1y=1 and throughs where y=1y=-1. The xx-coordinates are evenly spaced out such that the peaks are a horizontal distance of 180180 from each other and the troughs also have the same gap between them. The curve goes on forever in the positive and negative xx-directions but never goes higher than y=1y=1 nor lower than y=1y=-1


The key points you need to know are the coordinates of the peaks, troughs and the xx-intercepts. It repeats itself every 360360 on the xx-axis.​



Cosine graph

Maths; Graphs; KS4 Year 10; Trigonometric graphs - Higher


The graph of y=cos(x)y=\cos(x) is very similar to the sine graph above, but is shifted horizontally by 9090 units (when the xx-axis is in degrees). Hence the cosine graph (often referred to as the "cos" graph) intersects the yy-axis at (0,1)(0,1). Like the sine graph, the cosine graph repeats itself every 360360 degrees on the xx-axis.


Example 1

Give the coordinates of two points where y=cos(x)y=\cos(x) intersects the xx-axis.


Reading off the graph, you can see that two examples of xx-intercepts are (90,0)(90,0) and (90,0)(-90,0)


Note: The cosine graph is mirrored in the yy-axis.



Tangent graph

Maths; Graphs; KS4 Year 10; Trigonometric graphs - Higher


The graph of y=tan(x)y=\tan(x) is different to the other two trigonometric graphs shown here in that it has a different shape and it repeats itself every 180180 degrees rather than just every 360360 degrees. You still need to know key points on it: it passes through the xx-axis at the origin and again every 180180 degrees in the positive and negative xx-directions.


At x=90x=90 (and also at every jump of 180180), there is what is called an asymptote. This is a point on a curve where it gets closer and closer to a point, but never reaches it. So the graph of y=tan(x)y=\tan(x)​ has no value when x=90x=90, but very close to x=90x=90, it has a yy-value that is very big in the positive direction (if x<90x<90) or very big in the negative direction (if x>90x>90​). 


Example 2

Using a calculator, find tan(89.9)\tan(89.9) and also tan(90.1)\tan(90.1).What happens if you type tan(90)\tan(90)?


Typing these into a scientific calculator gives

tan(89.9)=572.96\tan(89.9)=\underline{572.96}


tan(90.1)=572.96\tan(90.1)=\underline{-572.96}​​

​​

each to two decimal places. Typing tan(90)\tan(90) gives a "math error" since there is no value for this - the graph never reaches x=90x=90. Another way of thinking of this is that coming from the left, y=tan(x)y=\tan(x) gets bigger as it approaches x=90x=90. It shoots off infinitely. But coming from the right, it shoots off to negative infinity. Since y=tan(x)y=\tan(x) cannot be infinity, nor negative infinity (and especially not both), it does not have a value.



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