Is it possible to sin without trig? That is a question that has plagued theologians for centuries. As evil as trigonometry is, modern science shows us that yes, it is possible to sin without trig. Here are some ways that I’ve come across.

1 – Slope Iteration Method

The above image uses 1024 samples from 0 to 2pi to aproximate sine iteratively using it’s slope. Red is true sin, green is the aproximation, and yellow is where they overlap.

This method comes from Game Programming Gems 8, which you can snag a copy of from amazon below if you are interested. It’s mentioned in chapter 6.1 A Practical DSP Radio Effect (which is a really cool read btw!).
Amazon: Game Programming Gems 8

This method uses calculus but is actually pretty straightforward and intuitive – and very surprising to me that it works so well!

The derivative of sin(x) is cos(x). That means, for any x you plug into sin, the slope of the function at that point is the cosine value of that same x.

In other words, sin(0) is 0, but it has a slope of cos(0) which is 1. Since slope is rise over run (change in y / change in x) that means that at sin(0), if you take an infinitely small step forward on x, you need to take the same sized step on y. That will get you to the next value of sine.

Let’s test that out!
sin(0) = 0 so we start at (0,0) on the graph.

If we try a step size of 0.1, our approximation is:
sin(0.1) = 0.1

The actual value according to google is 0.09983341664. so our error was about 0.0002. That is actually pretty close!

How about 0.25?
sin(0.25) = 0.25

The real value is 0.24740395925, so our error is about 0.003. We have about 10 times the error that we did at 0.1.

what if we try it with 0.5?
sin(0.5) = 0.5

The actual value is 0.4794255386, which leaves us with an error of about 0.02. Our error is 100 times as much as it was at 0.1. As you can see, the error increases as our step size gets larger.

If we wanted to, we could get the slope (cosine value) at the new x value and take another step. we could continue doing this to get our sine approximation, knowing that the smaller the step that we use, the more accurate our sine approximation would be.

We haven’t actually won anything at this point though, because we are just using cosine to take approximated steps through sine values. We are paying the cost of calculating cosine, so we might as well just calculate sine directly instead.

Well luckily, cosine has a similar relationship with sine; the derivative of cos(x) is -sin(x).

Now, we can use cosine values to step through sine values, and use those same sine values to step through cosine values.

Since we know that cos(0) = 1.0 and sin(0) = 0.0, we can start at an angle of 0 with those values and we can iteratively step through the values.

Here is a sample function in C++

// theta is the angle we want the sine value of.
// higher resolution means smaller step size AKA more accuracy but higher computational cost.
// I used a resolution value of 1024 in the image at the top of this section.
float SineApproximation (float theta, float resolution)
{
    // calculate the stepDelta for our angle.
    // resolution is the number of samples we calculate from 0 to 2pi radians
    const float TwoPi = 6.28318530718f;
    const float stepDelta = (TwoPi / resolution);

    // initialize our starting values
    float angle = 0.0;
    float vcos = 1.0;
    float vsin = 0.0;

    // while we are less than our desired angle
    while(angle < theta) {

        // calculate our step size on the y axis for our step size on the x axis.
        float vcosscaled = vcos * stepDelta;
        float vsinscaled = vsin * stepDelta;

        // take a step on the x axis
        angle += stepDelta;

        // take a step on the y axis
        vsin += vcosscaled;
        vcos -= vsinscaled;
    }

    // return the value we calculated
    return vsin;
}

Note that the higher the angle you use, the more the error rate accumulates. One way to help this would be to make sure that theta was between 0 and 2pi, or you could even just calculate between 0 and pi/2 and mirror / reflect the values for the other quadrants.

This function is quite a bit of work to calculate a single value of sine but it’s real power comes in the fact that it’s iterative. If you ever find yourself in a situation where you need progressive values of sine, and have some fixed angle step size through the sine values, this algorithm just needs to do a couple multiplies and adds to get to the next value of sine.

One great use of this could be in DSP / audio synthesis, for sine wave generation. Another good use could be in efficiently evaluating trigonometry based splines (a future topic I plan to make a post about!).

You can see this in action in this shadertoy or look below at the screenshots:
Shadertoy: Sin without Trig

64 Samples – Red is true sine, green is our approximation, and yellow is where they are the same

128 Samples

256 Samples

1024 Samples

2 – Taylor Series Method

Another way to calculate sine is by using an infinite Taylor series. Thanks to my friend Yuval for showing me this method.

You can get the Taylor series for sine by typing “series sin(x)” into wolfram alpha. You can see that here: Wolfram Alpha: series sin(x).

Wolfram alpha says the series is: x-x^3/6+x^5/120-x^7/5040+x^9/362880-x^11/39916800 ….

what this means is that if you plug a value in for x, you will get an approximation of sine for that x value. It’s an infinite series, but you can do as few or as many terms as you want to be able to trade off speed for accuracy.

For instance check out these graphs.

Google: graph y = x, y = sin(x)

Google: graph y = x-x^3/6, y = sin(x)

Google: graph y = x-x^3/6+x^5/120, y = sin(x)

Google: graph y = x-x^3/6+x^5/120-x^7/5040, y = sin(x)

Google: graph y = x-x^3/6+x^5/120-x^7/5040+x^9/362880, y = sin(x)

Google: graph y = x-x^3/6+x^5/120-x^7/5040+x^9/362880-x^11/39916800, y = sin(x)
(Note that I had to zoom out a bit to show where it became inaccurate)

When looking at these graphs, you’ll probably notice that very early on, the approximation is pretty good between -Pi/2 and + Pi/2. I leveraged that by only using those values (with modulus) and mirroring them to be able to get a sine value of any angle with more accuracy.

When using just x-x^3/6, there was an obvious problem at the top and bottom of the sine waves:

When i boosted the equation with another term, bringing it to x-x^3/6+x^5/120, my sine approximation was much better:

You can see this method in action in this shadertoy:
Shadertoy: Sin without Trig II

3 – Smoothstep Method

The third method may be my favorite, due to it’s sheer elegance and simplicity. Thanks to P_Malin on shadertoy.com for sharing this one with me.

There’s a function in graphics called “smoothstep” that is used to take the hard linear edge off of things, and give it a smoother, more organic feel. You can read more about it here: Wikipedia: Smoothstep.

BTW if you haven’t read the last post, I talk about how smooth step is really just a 1d bezier curve with specific control points (One Dimensional Bezier Curves). Also, smoothstep is just this function: y = (3x^2 – 2x^3).

Anyhow, if you have a triangle wave that has values from 0 to 1 on the y axis, and put it through a smoothstep, then scale it to -1 to +1 on the y axis, you get a pretty darn good sine approximation out.

Here is a step by step recipe:

Step 1 – Make a triangle wave that has values on the y axis from 0 to 1

Step 2 – Put that triangle wave through smoothstep (3x^2 – 2x^3)

Step 3 – Scale the result to having values from -1 to +1 on the axis.

That is pretty good isn’t it?

You can see this in action in this shadertoy (thanks to shadertoy’s Dave_Hoskins for some help with improving the code):
Shadertoy: Sin without Trig III

After I made that shadertoy, IQ, the creator of shadertoy who is an amazing programmer and an impressive “demoscene” guy, said that he experimented with removing the error from that technique to try to get a better sin/cos aproximation.

You can see that here: Shadertoy: Sincos approximation

Also, I recommend checking out IQ’s website. He has a lot of interesting topics on there: IQ’s Website

4 – CORDIC Mathematics

This fourth way is a method that cheaper calculators use to calculate trig functions, and other things as well.

I haven’t taken a whole lot of time to understand the details, but it looks like it’s using imaginary numbers to rotate vectors iteratively, doing a binary search across the search space to find the desired values.

The benefit of this technique is that it can be implemented with VERY little hardware support.

The number of logic gates for the implementation of a CORDIC is roughly comparable to the number required for a multiplier as both require combinations of shifts and additions.
Wikipedia: Coordinate Rotation Digital Computer

Did I miss any?

If you know of something that I’ve left out, post a comment, I’d love to hear about it!