Step by step
The question
Question: graph y = 2 cot 2x.
The x values are in degrees.
1 Why cot has gaps
cot x is cos x over sin x: tan turned upside down.
When sin x is 0, we would divide by 0.
That is never allowed: a gap.
sin x is 0 at x = 0, at 180°, and every 180° after that: on the x-axis of the unit circle.
Here the angle is 2x.
2x is 180° when x is 90°.
2 Find the period
Like tan, cot repeats every half turn: 180° over B.
B is 2: the period is 90°.
Careful!
Each piece of cot goes down from left to right.
tan goes up.
They are upside down of each other.
3 Make the key points
Between the gaps at 0 and 90°, three key points.
x goes a quarter period at a time: 22.5°.
cot at these points: 1, 0, −1.
Like cot 45°, cot 90° and cot 135°.
Each y is 2 times that.
4 Draw the graph
Draw the grid: a mark every 22.5°.
The gaps: dashed lines at x = 0 and every 90°.
Plot the key points.
Join them with a smooth curve, going down.
Near the dashed lines it shoots up and down.
The same piece again after each gap.
5 Check
Let's check one point from the table: x = 22.5°.
The inside is 45°, and cot of it is 1.
So y is 2, like in the table.
Right!
Let's look back.
First: why cot has gaps.
Then: find the period.
Then: make the key points.
Then: draw the graph.
Last: check.