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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)=572.96
tan(90.1)=−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)=572.96
tan(90.1)=−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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