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# Euler's sine wave

A sine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js. Created by Willy McAllister.

## Want to join the conversation?

• this is just calming for some reason lol
• Magnitude of the vector is smaller but the amplitude of produced sine wave is larger. How this happens?
• Did you notice the vector add happening? It's represented by the dotted lines. The red and green short vectors are added together to get the full height of the sine.
• I absolutely love this video. Dropping the student into a quiet lounge and just laying out the conclusions from all the lessons is genius!
• Watching this illustration , it looks like the result sin figure could also be generated by just e^jwt (without the e^-jwt and /2i) => sinwt = e^jwt (magnitude = 1)
What am I missing?
• Andrew is correct. The two conjugate exponential terms combine by vector addition to make the result fall on the imaginary axis. The 1/2 term is a scaling factor on each exponential to make the resulting sine come out with a magnitude of 1.
• I was wondering, can you represent the sine without the imaginary as: sin(wt) = (e^wt - e^-wt)/2?
• Sorry, no. If you drop the i out of the exponent the expression is two ordinary exponential terms. The one with the positive exponent, e^+wt, will rapidly become very large.
• I am in grade 6, do you think that I should learn this stuff?
• Sure! Enjoy browsing around this section with Euler's sine wave and cosine wave. But I wouldn't make this my beginning spot. Go to the introduction of Electrical Engineering and start there. I wish I had this to learn from when I started my engineering adventures in 6th grade.
• play at x2 speed it sounds different
• may i know what happens to the j in the denominator. eulers sine wave is defined as (e^jwt-e^-jwt) / 2j. im asking wht is the effect of this j in the numerator? please tell.
(1 vote)
• An important identity is 1/j = -j.
Any time you see a j in a denominator you can bring it up to the numerator as -j.
The j factor causes the moving yellow dot to travel up and down the imaginary axis.