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Chapter overview
Learning goals
Learning Goals
Maths
Summary
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.
The graph of $y=\sin(x)$ is pictured above. The $x$-axis displays degrees, the unit of angle. You can see that the graph passes through the origin and has peaks where $y=1$ and throughs where $y=-1$. The $x$-coordinates are evenly spaced out such that the peaks are a horizontal distance of $180$ from each other and the troughs also have the same gap between them. The curve goes on forever in the positive and negative $x$-directions but never goes higher than $y=1$ nor lower than $y=-1$.
The key points you need to know are the coordinates of the peaks, troughs and the $x$-intercepts. It repeats itself every $360$ on the $x$-axis.
The graph of $y=\cos(x)$ is very similar to the sine graph above, but is shifted horizontally by $90$ units (when the $x$-axis is in degrees). Hence the cosine graph (often referred to as the "cos" graph) intersects the $y$-axis at $(0,1)$. Like the sine graph, the cosine graph repeats itself every $360$ degrees on the $x$-axis.
Give the coordinates of two points where $y=\cos(x)$ intersects the $x$-axis.
Reading off the graph, you can see that two examples of $x$-intercepts are $(90,0)$ and $(-90,0)$.
Note: The cosine graph is mirrored in the $y$-axis.
The graph of $y=\tan(x)$ is different to the other two trigonometric graphs shown here in that it has a different shape and it repeats itself every $180$ degrees rather than just every $360$ degrees. You still need to know key points on it: it passes through the $x$-axis at the origin and again every $180$ degrees in the positive and negative $x$-directions.
At $x=90$ (and also at every jump of $180$), 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 $y=\tan(x)$ has no value when $x=90$, but very close to $x=90$, it has a $y$-value that is very big in the positive direction (if $x<90$) or very big in the negative direction (if $x>90$).
Using a calculator, find $\tan(89.9)$ and also $\tan(90.1)$.What happens if you type $\tan(90)$?
Typing these into a scientific calculator gives
$\tan(89.9)=\underline{572.96}$
$\tan(90.1)=\underline{-572.96}$
each to two decimal places. Typing $\tan(90)$ gives a "math error" since there is no value for this - the graph never reaches $x=90$. Another way of thinking of this is that coming from the left, $y=\tan(x)$ gets bigger as it approaches $x=90$. It shoots off infinitely. But coming from the right, it shoots off to negative infinity. Since $y=\tan(x)$ cannot be infinity, nor negative infinity (and especially not both), it does not have a value.
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.
The graph of $y=\sin(x)$ is pictured above. The $x$-axis displays degrees, the unit of angle. You can see that the graph passes through the origin and has peaks where $y=1$ and throughs where $y=-1$. The $x$-coordinates are evenly spaced out such that the peaks are a horizontal distance of $180$ from each other and the troughs also have the same gap between them. The curve goes on forever in the positive and negative $x$-directions but never goes higher than $y=1$ nor lower than $y=-1$.
The key points you need to know are the coordinates of the peaks, troughs and the $x$-intercepts. It repeats itself every $360$ on the $x$-axis.
The graph of $y=\cos(x)$ is very similar to the sine graph above, but is shifted horizontally by $90$ units (when the $x$-axis is in degrees). Hence the cosine graph (often referred to as the "cos" graph) intersects the $y$-axis at $(0,1)$. Like the sine graph, the cosine graph repeats itself every $360$ degrees on the $x$-axis.
Give the coordinates of two points where $y=\cos(x)$ intersects the $x$-axis.
Reading off the graph, you can see that two examples of $x$-intercepts are $(90,0)$ and $(-90,0)$.
Note: The cosine graph is mirrored in the $y$-axis.
The graph of $y=\tan(x)$ is different to the other two trigonometric graphs shown here in that it has a different shape and it repeats itself every $180$ degrees rather than just every $360$ degrees. You still need to know key points on it: it passes through the $x$-axis at the origin and again every $180$ degrees in the positive and negative $x$-directions.
At $x=90$ (and also at every jump of $180$), 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 $y=\tan(x)$ has no value when $x=90$, but very close to $x=90$, it has a $y$-value that is very big in the positive direction (if $x<90$) or very big in the negative direction (if $x>90$).
Using a calculator, find $\tan(89.9)$ and also $\tan(90.1)$.What happens if you type $\tan(90)$?
Typing these into a scientific calculator gives
$\tan(89.9)=\underline{572.96}$
$\tan(90.1)=\underline{-572.96}$
each to two decimal places. Typing $\tan(90)$ gives a "math error" since there is no value for this - the graph never reaches $x=90$. Another way of thinking of this is that coming from the left, $y=\tan(x)$ gets bigger as it approaches $x=90$. It shoots off infinitely. But coming from the right, it shoots off to negative infinity. Since $y=\tan(x)$ cannot be infinity, nor negative infinity (and especially not both), it does not have a value.
FAQs
Question: What is an asymptote?
Answer: This is a point on a curve where it gets closer and closer to a point, but never reaches it.
Question: How often do the sine and cosine graphs repeat themselves?
Answer: They repeats themselves every 360 degrees on the x-axis.
Question: What are the three key trigonometric graphs?
Answer: y=sin(x), y=cos(x) and y=tan(x).
Theory
Exercises
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