About Graph of Sine and Cosine
This page opens with y = sin(x) and y = cos(x) drawn from −2π to 2π. The two waves have the same shape, cos(x) being sin(x) shifted left by π/2, and they cross wherever the two values are equal. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The crossings repeat every π, at x = π/4 + kπ, where sin(x) = cos(x) = ±√2/2. In the window shown that is four points, and they are listed in the table under the graph. Add y = a*sin(x) to get a slider for the amplitude, or y = sin(b*x) for the frequency.
The calculator works in radians, so one full wave spans 2π ≈ 6.283 units on the x-axis.
How to use Graph of Sine and Cosine
- Type an expression in the first row, such as y = sin(x), or just sin(x). The graph updates as you type.
- Press Add expression for more rows. Use the coloured dot to change the colour and the eye to hide a row.
- Use a letter such as a, b or m in an expression and a slider for it appears under the list. Open Range to change its minimum, maximum and step.
- Drag the graph to move it, pinch or scroll to zoom, and use Fit or Square to reset the window. Hover or tap the graph to trace it.
- Read the Roots and intersections table, or the Table of values for any function.
- Press the share link button to copy a link to the graph, or use the download button under the graph for a PNG image.
Features
- A list of expressions: functions y = f(x), vertical lines x = c and points (a, b).
- A colour and a show or hide switch for every expression, with a typeset preview of what was understood.
- Automatic sliders for every free parameter, with editable minimum, maximum and step.
- Roots, y-intercepts and intersections between curves, found numerically, marked and listed.
- Trace by hovering or tapping, and a trace slider that reads every function at one x.
- Fit, Square, zoom in, zoom out and reset, with drag, pinch and wheel on the graph.
- A table of values for any function, with your own start and step.
- Linear and quadratic regression on a table of data points, with the equation and R².
- PNG export, and a share link that stores the graph in the URL hash so nothing is sent to a server.
- Presets that open real example graphs: sine and cosine, a parabola with sliders, a system of equations and two regressions.
Worked examples
Crossings of sin(x) and cos(x), −2π to 2π gives (-5.49779, 0.707107), (-2.35619, -0.707107), (0.785398, 0.707107), (3.92699, -0.707107)
x = π/4 + kπ, where both equal ±0.7071.
Roots of sin(x), −7 to 7 gives -6.28319, -3.14159, 0, 3.14159, 6.28319
Multiples of π.
Privacy: how it works
- The graph is computed and drawn in your browser. The share link keeps the expressions after the # sign of the address, which browsers never send to a server, so the people you share it with receive the graph from the link itself and Ulkaa never sees it.
Tips
- Press Fit after typing a function to bring the curve into view, and Square to give both axes the same scale so circles look round and slopes look true.
- To see how a parameter acts, put one curve on screen and drag its slider slowly. Open Range to widen the slider if the value you want is out of reach.
- On a phone the graph is at the top and the list scrolls below it, so you can edit and watch at the same time.
- Hide a row with the eye instead of deleting it to compare graphs quickly.
- The share link only contains the graph. Anyone with the link sees it exactly as you left it, with the same sliders and window.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your expressions are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your expressions are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
Where do sine and cosine intersect?
At x = π/4 + kπ for whole k. At those points both functions equal √2/2 or −√2/2, about 0.7071 or −0.7071.
How is cosine related to sine?
cos(x) = sin(x + π/2). The cosine wave is the sine wave shifted left by a quarter period.
More Graphing Calculator pages
- Graphing CalculatorPlot several functions at once, drag sliders, and read off roots and intersections.
- Parabola GrapherDrag a, b and c and watch the parabola, its roots and its vertex change.
- Linear Equation GrapherDrag the slope m and the intercept b and watch the line move.
- Cubic Function GrapherA cubic with its three roots, two turning points and a horizontal line cutting it.
- Tangent Function GrapherSee tan(x) shoot to infinity at each asymptote and cross zero at every multiple of π.
- Absolute Value Graph CalculatorThe V-shaped graph of abs(x − 2) + 1 and the horizontal line y = 4.
- Exponential Function GrapherCompare 2^x and e^x: both pass through (0, 1) and then grow at different speeds.
- Logarithm Graph CalculatorThe natural log ln(x) and the base-10 log(x) on the same axes.
- System of Equations GrapherGraph two equations and read the solution where the lines cross.
- Linear Regression GrapherEnter x and y pairs, see them as points and fit the best straight line.
- Quadratic Regression GrapherFit y = ax² + bx + c to seven points and see how well a parabola describes them.