How to Graph Sine and Cosine Functions

Reviewed by Joseph Meyer

Last Updated: June 2, 2024

The sine and cosine functions appear all over math in trigonometry, pre-calculus, and even calculus. Understanding how to create and draw these functions is essential to these classes, and to nearly anyone working in a scientific field. This article will teach you how to graph the sine and cosine functions by hand, and how each variable in the standard equations transform the shape, size, and direction of the graphs.

Part 1

Part 1 of 3:

Graphing the Basic Equations

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1
Draw a coordinate plane.
- For a sine or cosine graph, simply go from 0 to 2π on the x-axis, and -1 to 1 on the y-axis, intersecting at the origin (0, 0).
- Both $y=\sin(x)$ and $y=\cos(x)$ repeat the same shape from negative infinity to positive infinity on the x-axis (you'll generally only graph a portion of it).
- Use the basic equations as given: $y=\sin(x)$ and $y=cos(x)$
2
Graph the basic form of $y=\sin(x)$ . Plot and connect the points (0, 0), (π/2, 1), (π, 0), and (3π/2, -1) with a continuous curve.
- Both $y=\sin(x)$ and $y=\cos(x)$ never go past -1 or 1 on the y-axis.
- Since you are only hand-drawing your graphs , there is no precise scale, but it must be accurate at certain points.
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3
Graph the basic form of $y=cos(x)$ . Plot and connect the points (0, 1), (π/2, 0), (π, -1), and (3π/2, 0) with a continuous curve.
- It may be helpful to use two separate colors to distinguish between sine and cosine.
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Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

Develop strong graphing skills. Drawing graphs by hand will help you develop foundational graphing skills, especially in understanding scales and axes. This will build a strong base for you to use helpful online tools to visualize complex relationships, perform calculations, and prepare for standardized tests.

Part 2

Part 2 of 3:

Graphing Different Sine Equations

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1
Use the standard equation to define your variables. $y=A\sin(Bx+C)+D$
- Find your values of A, B, C, and D.
- Note that in the basic equation for sine, A = 1, B = 1, C = 0, and D = 0.
2
Calculate the period.
- Divide your period on the x-axis into four sections that are equal distances apart, just like in the basic equations. The y-values will still alternate from 0, 1, 0, and -1 just like in the basic equation.
- ${\text{Period}}={\frac {2\pi }{B}}$
3
Calculate the amplitude.
- ${\text{Amplitude}}=A$
- Multiply the y-values you have by A, and graph these new points.
- If A is negative, the graph will flip over the x-axis. This is called a reflection.
4
Calculate the phase shift.
- ${\text{Phase shift}}={\frac {C}{B}}$
- This will move the graph to the left or right.
- For each x-value in the period, move the x-value to the left by C/B if C/B is negative, or move each x-value to the right by C/B if C/B is positive.
5
Calculate the vertical shift.
- ${\text{Vertical shift}}=D$
- For each y-value, move the y-value up by D if D is positive, or move the y-value down if D is negative.
After each transformation has been applied, your graph is finished!

I.M. Gelfand, Mathematician

To accurately sketch sine and cosine curves, first mark the key points where y=0 or y=±1 based on their values at special angles on the unit circle. Connect those anchor points smoothly, minding the amplitude, period, phase shifts or compressions introduced by the function's coefficients. The intrinsic symmetry and cyclic nature of the waves emerges beautifully when following this methodical approach.

Part 3

Part 3 of 3:

Graphing Different Cosine Equations

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1
Use the standard equation to define your variables. $y=A\cos(Bx+C)+D$
- Find your values of A, B, C, and D.
- Note that in the basic equation for cosine, A = 1, B = 1, C = 0, and D = 0.
2
Calculate the period.
- Divide your period on the x-axis into four sections that are equal distances apart, just like in the basic equations. The y-values will still alternate from 1, 0, -1, and 0 just like in the basic equation.
- ${\text{Period}}={\frac {2\pi }{B}}$
3
Calculate the amplitude.
- ${\text{Amplitude}}=A$
- Multiply the y-values you have by A, and graph these new points.
- If A is negative, the graph will flip over the x-axis. This is called a reflection.
4
Calculate the phase shift.
- ${\text{Phase shift}}={\frac {C}{B}}$
- This will move the graph to the left or right.
- For each x-value in the period, move the x-value to the left by C/B if C/B is negative, or move each x-value to the right by C/B if C/B is positive.
5
Calculate the vertical shift.
- ${\text{Vertical shift}}=D$
- This will move the graph up or down.
- For each y-value, move the y-value up by D if D is positive, or move the y-value down if D is negative.
After each transformation has been applied, your graph is finished!

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Tips

Keep track of your variables as you go through each step - write them down next to each graph of a transformation, or use a table to change each point as you apply a new transformation.

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Use a graphing calculator or software to see how the graph changes with each new variable.

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Warnings

It is important to calculate the amplitude and period first, because they will make it easier to apply the phase shift and vertical shift later on.

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Any negative values for a variable will essentially do the opposite of what a positive variable would do.

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Things You'll Need

Pencil or pen to draw
Different colors to display the different shapes of the graph (optional)
Graphing calculator or software to check your work (optional)

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How to Graph Sine and Cosine Functions

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Graphing the Basic Equations

Graphing Different Sine Equations

Graphing Different Cosine Equations

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