Complex Variables Trigonometric Identity Proof sin^2(z) + cos^2(z) = 1 | Summary and Q&A

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January 21, 2020
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The Math Sorcerer
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Complex Variables Trigonometric Identity Proof sin^2(z) + cos^2(z) = 1

TL;DR

This video provides a proof of the equation sine squared of Z plus cosine squared of Z equals 1 for complex numbers.

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Key Insights

  • 👨‍💼 The definitions of sine and cosine for complex numbers involve exponential functions.
  • 😑 The proof uses algebraic properties and shortcuts to simplify the expression.
  • 🤘 Cancelations occur when terms with opposite signs are combined.
  • 🍵 The proof demonstrates how to handle complex numbers in trigonometric equations.
  • 🧑‍🏭 It is important to distinguish between sine and cosine definitions due to the presence of an additional factor of I.
  • 😒 Mathematicians often use shorthand methods to simplify calculations.
  • 😑 Distributing a negative sign and finding a common denominator can be useful in rearranging expressions.

Transcript

in this video we're going to prove that sine squared of Z plus cosine squared of Z is equal to 1 in this problem Z is a complex number so this proof will be a little bit different than the proof you would see in a trigonometry class let's go ahead and go through it carefully so proof so first it's really important to know what the actual definition... Read More

Questions & Answers

Q: What is the definition of cosine for complex numbers?

The cosine of a complex number Z is given by e to the I Z plus e to the negative I Z all over 2.

Q: How is the definition of sine for complex numbers different from cosine?

The definition of sine for complex numbers is e to the I Z minus e to the negative I Z all over 2, and it involves an additional factor of I.

Q: How does the proof simplify the expression sine squared Z plus cosine squared Z?

The proof applies algebraic properties, such as the square of a sum, to expand and simplify the expression step by step.

Q: How does the proof conclude?

By rearranging and canceling terms, the proof shows that the expression simplifies to 1, proving the equation for complex numbers.

Summary & Key Takeaways

  • The video presents the definitions of sine and cosine for complex numbers.

  • The proof starts by writing down the left hand side of the equation and replacing sine and cosine with their complex number definitions.

  • Math operations are applied to simplify the expression and eventually show that the left hand side equals 1.

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