Saturday, March 8, 2014

Rediscovering Trigonometry Part 2: Double Angle Formulas

Click here to see part one of this four week series.

Last week, we developed some formulas that could calculate the sine, cosine, and tangent of the sum and difference of two angles. This proves useful when trying to calculate the exact sines/cosines/tangents of angles that are not in special right triangles (45-45-90, 30-60-90, 18-72-90).

What if we wanted to find the exact values for double a certain angle. Let's say we know the following:

sin18 = (√(5) - 1)/4
cos18 = (√(10 + 2√(5)))/4

How could we calculate the sine of 36°? We know the sine angle addition formula from last week, but it would be much easier to have a generalized version of this. Let's take a look at it.

sin(α + β) = sinα cosβ + cosα sinβ

What if we were finding the sine of 2α? Let's see what the formula would tell us:

sin(2α) = sin(α + α) = sinα cosα + cosα sinα = 2sinα cosα

So with some pretty simple computations, we get the formula:

sin(2α) = 2sinα cosα

That's a pretty simple formula. To find the sine of 36°, we could just do some easy manipulation with the values from up top.

sin36 = sin(2 • 18) = 2 • ((√(5) - 1)/4) • ((√(10 + 2√(5)))/4) = (√(10 - 2√(5)))/4

This sort of computation is clearly very useful when you are studying trigonometry. It wouldn't be practical in the sense that we use it in our day-to-day lives, but I think it is clear how useful this is to mathematicians and astronomers. I also find it really cool that these seemingly random irrational values can be derived exactly using just some basic mathematics.

Let's create a cosine formula. We know from last week that:

cos(α + β) = cosα cosβ – sinα sinβ

Substituting α in for β gives:

cos(2α) = cos(α + α) = cosα cosα – sinα sinα = cos2α – sin2α

This is the formula that naturally comes out:

cos(2α) = cos2α – sin2α

Knowing that sin2α + cos2α = 1, this can be rewritten in a few different ways:

cos(2α) = cos2α – sin2α
cos(2α) = 2cos2α – 1
cos(2α) = 1 – 2sin2α

This is extremely convenient, as the problem can be made much easier depending on what information you have. If you only know the sine of the angle, the third formula will work. If you only know the cosine, the second formula will work. There are also times where the first formula might be most convenient.

As you can see, it is pretty simple to do the work to come up with the double angle formulas, probably even easier than applying them in many cases. I encourage you to do the same process as we have done for sines and cosines to generate one for tangents, using the tangent formula we found last week. You will get a very pretty result.

Also, it can be fun to play around with these and find the exact sines and cosines of various angles. You will also see that there are many ways to write these different values. This is because they are irrational numbers. You could find the sine of 105° by doing sin(45 + 60), sin(90 + 15), sin(180 - 75), or many other variations. These could very well give different looking answers, but if the math was correct, the actual results will be equal.

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