Pre-Calculus
Lesson 22:
Graphs of the Sine and Cosine Functions
1. Def: A periodic function is a function 𝑓 such that
𝑓(𝑥) = 𝑓(𝑥 + 𝑛𝑝),
for every real number 𝑥 in the domain of 𝑓, every integer 𝑛, and some positive real number 𝑝.
The smallest possible positive value of 𝑝 is the period of the function.
2. The Sine Wave:
Domain: (−∞, ∞)




Range: [−1, 1]
The graph is continuous over its entire domain.
Its 𝑥-intercepts are of the form 𝑛𝜋, where 𝑛 is an integer.
Its period is 2𝜋.
The graph is symmetric with respect to the origin, so the function is an odd function.
For all 𝑥 in the domain, sin(−𝑥) = − sin 𝑥.
3. The Cosine Wave:
Domain: (−∞, ∞)
Range: [−1, 1]

The graph is continuous over its entire domain.

The 𝑥-intercepts are of the form (2𝑛 + 1) 2 , where 𝑛 is an integer.


Its period is 2𝜋.
The graph is symmetric with respect to the 𝑦-axis, so the function is an even function.
For all 𝑥 in the domain, cos(−𝑥) = cos 𝑥.
𝜋
4. Graphing 𝑦 = 𝑎 sin 𝑥
Ex. - Graph 𝑦 = 2 sin 𝑥
5. Def: The amplitude of a periodic function is half the distance between the maximum and
minimum values. Therefore, the graph of 𝑦 = 𝑎 sin 𝑥 or 𝑦 = 𝑎 cos 𝑥, with 𝑎 ≠ 0, will have the
same shape as the graph of 𝑦 = sin 𝑥 or 𝑦 = cos 𝑥, respectively, except with range [−|𝑎|, |𝑎|].
The amplitude is |𝑎|.
6. Graphing 𝑦 = sin 𝑏𝑥
Ex. - Graph 𝑦 = sin 2𝑥
For 𝑏 > 0, the graph of 𝑦 = sin 𝑏𝑥 will resemble that of 𝑦 = sin 𝑥, but with period
graph of 𝑦 = cos 𝑏𝑥 will resemble that of 𝑦 = cos 𝑥, but with
2𝜋
.
𝑏
Also, the
2𝜋
period 𝑏 .
7. Graphing 𝑦 = cos 𝑏𝑥
2
Ex. - Graph 𝑦 = cos 3 𝑥 over one period.
First, we find the period:
2𝜋 2𝜋
3
=
= 2𝜋 ∙ = 3𝜋
2
𝑏
2
3
Next, we split the interval [0, 3𝜋] into four equal parts to get the minimum points, maximum
points, and 𝑥-intercepts.
0,
3𝜋 3𝜋 9𝜋
, , , 3𝜋
4 2 4
Then we use these values to obtain the key points for one period.
3𝜋
3𝜋
9𝜋
X
0
4
2
4
2
𝜋
3𝜋
0
𝜋
x
3
2
2
2
1
0
−1
0
cos 𝑥
3
3𝜋
2𝜋
1
8. Guidelines for Sketching Graphs of Sine and Cosine Functions
To graph 𝑦 = 𝑎 sin 𝑏𝑥 or 𝑦 = 𝑎 cos 𝑏𝑥, with 𝑏 > 0, follow these steps.
Step 1: Find the period,
2𝜋
.
𝑏
Start at 0 on the 𝑥-axis, and lay off a distance of
2𝜋
.
𝑏
Step 2: Divide the interval into four equal parts.
Step 3: Evaluate the function for each of the five 𝑥-values resulting from Step 2. The points will
be maximum points, minimum points, and 𝑥-intercepts.
Step 4: Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude
|𝑎|.
Step 5: Draw the graph over additional periods, to the right and to the left, as needed.
9. Graphing 𝑦 = 𝑎 sin 𝑏𝑥
Ex. - 𝑦 = −2 sin 3𝑥 over one period
Step 1:
2𝜋
𝑏
=
2𝜋
3
[0,
2𝜋
]
3
𝜋 𝜋 𝜋 2𝜋
3
Step 2: 0, 6 , 3 , 2 ,
Step 3:
𝑥
0
3𝑥
0
sin 3𝑥
−2 sin 3𝑥
0
0
𝜋
6
𝜋
2
1
−2
𝜋
3
𝜋
0
0
𝜋
2
3𝜋
2
−1
2
2𝜋
3
2𝜋
0
0
Steps 4 and 5: on board
10. Def: In trig, horizontal translations are called phase shifts.
11. Graphing 𝑦 = sin(𝑥 − 𝑑)
𝜋
Ex. - Graph 𝑦 = sin (𝑥 − 3 )
Method 1:
𝜋
3
For the argument 𝑥 − to result in all possible values throughout one period, it must take on all
values between 0 and 2𝜋, inclusive. Therefore, to find an interval of one period, we solve the
three-part inequality
𝜋
0 ≤ 𝑥 − ≤ 2𝜋
3
𝜋
7𝜋
≤𝑥≤
3
3
𝜋 7𝜋
𝜋 5𝜋 4𝜋 11𝜋 7𝜋
Divide the interval [ 3 , 3 ] into four equal parts: 3 , 6 , 3 , 6 , 3
𝜋
3
0
𝑥
𝑥−
𝜋
3
𝜋
sin (𝑥 − )
3
0
4𝜋
3
𝜋
5𝜋
6
𝜋
2
1
0
11𝜋
6
3𝜋
2
−1
7𝜋
3
2𝜋
0
Plot points and connect with wave.
Method 2:
𝜋
The argument 𝑥 − 3 indicates a phase shift to the right compared to the graph of 𝑦 = sin 𝑥. So
𝜋
we simply shift the graph of 𝑦 = sin 𝑥 to the right by 3 .
12. Vertical Translations cause the graph to move up or down.
13. Graphing 𝑦 = 𝑐 + 𝑎 cos 𝑏𝑥
Ex. - Graph 𝑦 = 3 − 2 cos 3𝑥
2𝜋 2𝜋
=
𝑏
3
𝜋 𝜋 𝜋 2𝜋
0, , , ,
6 3 2 3
𝑥
0
3𝑥
0
cos 3𝑥
2 cos 3𝑥
3 − 2 cos 3𝑥
1
2
1
𝜋
6
𝜋
2
0
0
3
𝜋
3
𝜋
−1
−2
5
𝜋
2
3𝜋
2
0
0
3
2𝜋
3
2𝜋
1
2
1
Plot points and connect with sinusoidal curve.
14. Combinations of Translations of the form
𝑦 = 𝑐 + 𝑎 sin 𝑏(𝑥 − 𝑑)
or
𝑦 = 𝑐 + 𝑎 cos 𝑏(𝑥 − 𝑑)
with 𝑏 > 0, can be graphed according to the following guidelines:
Step 1: Find an interval whose length is one period
2𝜋
𝑏
by solving the three-part inequality 0 ≤
𝑏(𝑥 − 𝑑) ≤ 2𝜋.
Step 2: Divide the interval into four equal parts.
Step 3: Evaluate the function for each of the five 𝑥-values resulting from Step 2. The points will
be maximum points, minimum points, and points that intersect the line 𝑦 = 𝑐 (“middle” points
of the wave).
Step 4: Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude
|𝑎|.
Step 5: Draw the graph over additional periods, to the right and to the left, as needed.
15. Graphing 𝑦 = 𝑐 + 𝑎 sin 𝑏(𝑥 − 𝑑)
Ex. - Graph 𝑦 = −1 + 2 sin(4𝑥 + 𝜋)
First, rewrite the formula to match the general form:
𝜋
𝑦 = −1 + 2 sin 4 (𝑥 + )
4
Next, find an interval whose length is one period.
𝜋
0 ≤ 4 (𝑥 + ) ≤ 2𝜋
4
𝜋 𝜋
0≤𝑥+ ≤
4 2
𝜋
𝜋
− ≤𝑥≤
4
4
𝜋 𝜋
Step 2: Divide the interval [− 4 , 4 ] into four equal parts:
𝜋 𝜋 𝜋 𝜋
− , − , 0, ,
4
8
8 4
Step 3:
𝜋
𝜋
𝑥
0
−
−
4
8
𝜋
𝜋
𝜋
0
𝑥+
4
8
4
𝜋
𝜋
0
𝜋
4 (𝑥 + )
4
2
𝜋
0
1
0
sin 4 (𝑥 + )
4
𝜋
0
2
0
2 sin 4 (𝑥 + )
4
𝜋
−1
1
−1
−1 + 2 sin 4 (𝑥 + )
4
Steps 4 and 5: Plot points and connect with sinusoidal curve.
HW: #1 - 19 odd
#23 - 51 odd
𝜋
8
3𝜋
8
3𝜋
2
−1
𝜋
4
𝜋
2
2𝜋
−2
0
−3
−1
0