The sine graph is the height of the circling point.
Five key points per period: zero, maximum, zero, minimum, zero. Plot those and join them smoothly and the shape is right.
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- Graphs of the Circular Functions
Graphs of the Circular Functions
Unroll the unit circle and the y-coordinate traces a sine wave.
Choose the period
What is the period of y = tan x?
Sine and cosine repeat every 2π, but tangent repeats every π. Half a turn sends the point on the unit circle to the exact opposite side, where x and y both change sign — so their ratio is unchanged. C is the period of sine and cosine, which is the natural but wrong assumption here.
Cosine is the same wave, started at the top.
y = sin x
starts at 0, rising
(0, 0)
y = cos x
starts at 1
(0, 1)
relationship
cos x = sin(x + π2)
the sine wave slid a quarter period (π2) to the LEFT
Both have period 2π, range −1 to 1, and the same shape. The only difference is where the wave is caught at x = 0, which is why a shift turns one into the other.
Tangent is different in every respect.
period
π, not 2π
half as long
range
all real numbers
no maximum or minimum
asymptotes
at x = π2, 3π2, …
wherever cos x = 0
Tangent has no amplitude
Amplitude measures half the distance between a maximum and a minimum, and tangent has neither — it runs off to infinity between every pair of asymptotes. Asking for the amplitude of a tangent graph is asking for something that does not exist.
The asymptotes come from dividing by zero.
tan x = sin xcos x
a quotient
at x = π2, cos x = 0
the bottom vanishes
sin(π2) = 1, not 0
so it is not a hole
a vertical asymptote
same test as a rational function
Exactly the rule from rational functions: bottom zero and top not zero means a wall. Nothing new is being learned, only applied to a new pair of functions.
Variations: cosine's key points, tangent's zeros, and negative x.
Reading the three graphs, in the places students most often misread them.
The five key points of y = cos x from 0 to 2π
x = 0, π2, π, 3π2, 2π
a quarter period apart
y = 1, 0, −1, 0, 1
maximum, zero, minimum, zero, maximum
(0, 1), (π2, 0), (π, −1), (3π2, 0), (2π, 1)
join smoothly
The same five x-values as sine; only the heights move along by one place.
The zeros of y = tan x from 0 to 2π
tan x = 0 where sin x = 0
the TOP is zero, the bottom is not
x = 0, π, 2π
three zeros
asymptotes at x = π2, 3π2
where cos x = 0 instead
Zeros come from the top of sin xcos x, asymptotes from the bottom. Mixing them up swaps the two lists.
Negative x: sin(−π2) and cos(−π)
sine is symmetric about the origin: sin(−x) = −sin x
an odd function
sin(−π2) = −sin(π2) = −1
a minimum, left of the y-axis
cosine is symmetric about the y-axis: cos(−x) = cos x
an even function
cos(−π) = cos π = −1
the same as on the right
Extend the graphs to the left of the y-axis by reflecting: cosine as a mirror image, sine turned upside down too.
Three facts describe any of these graphs.
| period | range | asymptotes | |
|---|---|---|---|
| sin x | 2π | −1 to 1 | none |
| cos x | 2π | −1 to 1 | none |
| tan x | π | all reals | where cos x = 0 |
Sine and cosine are bounded: no matter how large x becomes, the output stays between −1 and 1. That is what makes them useful for anything that oscillates without growing — tides, sound, alternating current.
On the exam
Tests ask for the period, range, amplitude or asymptotes of sin x, cos x or tan x, and like to offer tangent an amplitude or a period of 2π. The fastest route: keep the three-row table in mind — 2π, 2π, π for the periods; −1 to 1, −1 to 1, all reals for the ranges. A 10-second check: picture the point on the unit circle; its height is sin x, its x-coordinate is cos x, and tangent fails where the x-coordinate is 0.
Now you try
Practice problem
Find the range
What is the range of y = cos x?
Cosine is the x-coordinate of a point on a circle of radius 1, so it can never leave −1 ≤ y ≤ 1, and it reaches both ends. B forgets the negative half — cos π = −1. A is the range of TANGENT.
Hint: picture the point going round the unit circle. How far left and right does it reach? That is the range of cosine.