{"id":1158,"date":"2014-10-08T09:57:38","date_gmt":"2014-10-08T13:57:38","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/?p=1158"},"modified":"2019-11-01T11:22:17","modified_gmt":"2019-11-01T15:22:17","slug":"eulers-formula-and-trig-identities","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2014\/10\/08\/eulers-formula-and-trig-identities\/","title":{"rendered":"Euler’s formula and trig identities"},"content":{"rendered":"\r\n

I'm terrible at rote memorization.<\/p>

I thought about that recently when I happened to see a plot (I think it was in a documentation example) that compared $sin(\\theta)$, $cos(\\theta)$, and $sin(\\theta) cos(\\theta)$.<\/p>

theta = linspace(0,6*pi,500);\r\nplot(theta,sin(theta))\r\nhold on<\/span>\r\nplot(theta,cos(theta))\r\nplot(theta,sin(theta) .* cos(theta))\r\nhold off<\/span>\r\nlegend({'sin(\\theta)'<\/span>,'cos(\\theta)'<\/span>,'sin(\\theta) cos(\\theta)'<\/span>})\r\n<\/pre>\"\" 
Hmm, I thought. I guess $sin(\\theta) cos(\\theta)$ is just a sinusoid with twice the frequency and half the amplitude.<\/p>
How is that related to memorization, you might ask? Well, I always had a very hard time remembering trig identities in high school and college. But, as an electrical engineering student, I had the following formula engraved on my brain cells:<\/p>
$$e^{j\\theta} = cos(\\theta) + j sin(\\theta)$$<\/p>
(Note that I am following the proud electrical engineering tradition of using j<\/i> instead of i<\/i> as the imaginary unit. Because i<\/i> obviously stands for current.)<\/p>
This is called Euler's formula, and it's used all over the place in electrical engineering. I did not know at the time that Richard Feynman called it \"the most remarkable formula in mathematics.\"<\/p>
Here are a couple of closely related formulas:<\/p>
$$cos(\\theta) = \\frac{1}{2} (e^{j\\theta} + e^{-j\\theta})$$<\/p>
$$sin(\\theta) = \\frac{1}{2j} (e^{j\\theta} - e^{-j\\theta})$$<\/p>
At some point during my undergraduate days, it dawned on me that I could derive many of the common trig identities directly from these formulas. Here's how it goes for $sin(\\theta) cos(\\theta)$:<\/p>
$$\r\nsin(\\theta) cos(\\theta) = \\frac{1}{2j} (e^{j\\theta} - e^{-j\\theta})\r\n\\frac{1}{2} (e^{j\\theta} + e^{-j\\theta})\r\n$$<\/p>
$$\r\nsin(\\theta) cos(\\theta) = \\frac{1}{4j} (e^{j2\\theta} - e^{-j2\\theta} + 1 - 1)\r\n$$<\/p>
$$\r\nsin(\\theta) cos(\\theta) = \\frac{1}{2} \\frac{1}{2j} (e^{j2\\theta} -\r\ne^{-j2\\theta})\r\n$$<\/p>
$$\r\nsin(\\theta) cos(\\theta) = \\frac{1}{2} sin(2\\theta)\r\n$$<\/p>
Presto! Like magic.<\/p>
I used to think that I must be kind of weird to prefer this manipulation of Euler's formula to simply memorizing the trig identities. But while writing this blog today, I noticed that the Wikipedia article on Euler's formula<\/a> describes this concept exactly. \"Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components.\"<\/p>
OK!<\/p>
PS. In my plot creation code above, I exploited a subtle improvement in MATLAB R2014b graphics that customers have been requesting for many years. Can you spot it?<\/p>