Graphs of Sine and Cosine Functions
In past sections we have seen how to evaluate trigonometric functions using the unit circle. Now
I want to look at the graphs of sine and cosine on the coordinate plane.
The Sine Function: 𝒚 = 𝒔𝒊𝒏𝒙
Domain: (−∞, ∞)
Range: [−1,1]
Start by making a table with values that are found from the unit circle:
𝒙
0
𝒚
0
𝜋
𝜋
𝜋
𝜋
2𝜋
3𝜋
5𝜋
6
4
3
2
3
4
6
1
√2
2
√3
2
1
√3
2
√2
2
1
2
2
𝜋
0
7𝜋
5𝜋
4𝜋
3𝜋
5𝜋
7𝜋
11𝜋
6
4
3
2
3
4
6
1
1
− 2 − √2 − √3 −1 − √3 − √2 − 2
2
2
2
2
Now we can plot these points on a coordinate plane:
Connect the points to obtain the sine curve:
Notice that the graph keeps going. Since the sine function is 2𝜋 periodic, the curve keeps
repeating itself off to infinity. Every x interval of length 2𝜋 will contain one full sine wave.
Also notice that the curve fluctuates between -1 and 1 since the unit circle has a radius of 1.
© Leslie Loy, School of Mathematical & Statistical Sciences – Arizona State University
2𝜋
0
The Cosine Function: 𝒚 = 𝒄𝒐𝒔𝒙
Domain: (−∞, ∞)
Range: [−1,1]
Start by making a table with values that are found from the unit circle:
𝒙
0
𝒚
1
𝜋
𝜋
𝜋
𝜋
2𝜋
3𝜋
5𝜋
6
4
3
2
3
4
6
√3
2
√2
2
1
0
− 2 − √2 − √3 −1 − √3 − √2 − 2
2
𝜋
7𝜋
5𝜋
4𝜋
3𝜋
5𝜋
7𝜋
11𝜋
6
4
3
2
3
4
6
1
√2
2
√3
2
1
1
2
2
2
2
0
2
2𝜋
1
Now we can plot these points on a coordinate plane:
Connect the points to obtain the cosine curve:
Notice that the graph keeps going, just as we saw with the graph of sine. Since the cosine
function is also 2𝜋 periodic, the curve keeps repeating itself off to infinity. Every x interval of
length 2𝜋 will contain one full cosine wave.
𝜋
These curves look very similar! If we take the graph of sine and shift it to the left by 2 , we get
𝜋
the graph of cosine. Think about transformations. A horizontal shift left by 2 means we would
𝜋
𝜋
need to add 2 on the inside of the sine function, which gives us: cos(𝑥) = sin⁡ (𝑥 + 2 ⁡)
Try graphing these two functions on the calculator to verify that this is true. (don’t forget radian mode!)
© Leslie Loy, School of Mathematical & Statistical Sciences – Arizona State University
When graphing sine and cosine, we can perform transformations on the basic graphs. The names
of the transformations are a little different, but the same concepts apply.
Example 1: Graph 𝑦 = 4cos⁡(𝑥).
This is a transformation we should all be familiar with, a vertical stretch by a factor of 4.
In terms of trig functions, this is called the amplitude.
1
Example 2: Graph 𝑦 = − 2 sin⁡(𝑥)
This is another example of a vertical transformation, this time a shrink. But wait! There is
a second transformation, an x-axis reflection because of the negative on the outside of the
1
function. The amplitude is 2 and there is an x-axis reflection.
© Leslie Loy, School of Mathematical & Statistical Sciences – Arizona State University
Example 3: Graph 𝑦 = 3cos⁡(2𝑥)
There are two transformations affecting the graph of 𝑦 = cos⁡(𝑥), a vertical stretch by 3
(so the amplitude is 3) and a horizontal shrink by 2. This will change the period, which is
how often the wave repeats. Remember, the period of the basic sine and cosine graphs is
2𝜋. Since this graph is shrunk by a factor of 2, the wave repeats in half the distance,
which is 𝜋.
To find the new period, divide 2𝜋 by
the number multiplied by x.
In this case
2𝜋
2
=𝜋.
There is one full wave on [0, 𝜋],
another full wave on [𝜋, 2𝜋] and this
continues infinitely.
𝜋
Example 4: Graph 𝑦 = sin (𝑥 − 4 )
𝜋
Here we have a horizontal shift right by 4 . In trigonometry, we call this a phase shift.
Since we are shifting right, the phase shift is positive.
𝜋
4
𝜋
Black: 𝑦 = sin (𝑥 − 4 )
Red:
𝑦 = sin(𝑥)
Notice: the distance
between the maximums
𝜋
is 4 .
© Leslie Loy, School of Mathematical & Statistical Sciences – Arizona State University
The general equation can be written as:
Amplitude:
|A|
𝒚 = 𝑨 𝐬𝐢𝐧(𝑩𝒙 − 𝑪) or 𝒚 = 𝑨 𝐜𝐨𝐬(𝑩𝒙 − 𝑪)
The distance between the x-axis and any maximum or minimum point.
(If 𝐴 < 0, there is also a reflection about the x-axis.)
Period:
𝟐𝝅
Phase Shift:
𝑪
The length it takes for one full wave to complete.
𝑩
Equivalent to the horizontal shift.
𝑩
(A positive phase shift means a shift right, and
a negative phase shift means a shift to the left.)
𝜋
Example 5: Find the amplitude, period and phase shift for the function 𝑦 = 2 cos (𝜋𝑥 + 2 ).
𝜋
Start by identifying A, B and C: 𝐴 = 2, 𝐵 = 𝜋, 𝐶 = − 2
Amplitude: |𝐴| = |2| = ⁡⁡⁡2
2𝜋
2𝜋
Period: 𝐵 = 𝜋 = ⁡⁡⁡2
𝐶
−
𝜋
𝜋
1
1
Phase Shift: 𝐵 = 𝜋2 = − 2 ∙ 𝜋 = ⁡⁡⁡ − 2
(remember this is a shift left so the phase shift is negative)
When graphing on the calculator, make
sure you’re in radian mode.
Next, set a good window. Always graph at
least one full wave, so look at the period.
If I choose 𝑥𝑚𝑖𝑛 = 0, then 𝑥𝑚𝑎𝑥 should be
at least 2 since the period is 2. I ended up
setting 𝑥𝑚𝑎𝑥 = 5, which is slightly larger
3𝜋
than 2 , so I could see more than two full
waves.
Setting the 𝑦 values in the window is
usually easier because you can look at the
amplitude. In this case set 𝑦𝑚𝑖𝑛 = −2 and
𝑦𝑚𝑎𝑥 = 2.
© Leslie Loy, School of Mathematical & Statistical Sciences – Arizona State University