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hartnn Group Title

Aaah! new group for trigonometry :) Let me start this by asking to expand sin 5x in powers of sin x. Only for those who have started learning trigonometry and never tried this. Those who already know how to, can give other methods, not discussed here, later.

  • 2 years ago
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  1. lgbasallote Group Title
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    using the new groups is fun, isn't it?

    • 2 years ago
  2. hartnn Group Title
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    its just the beginning...... Another similar exercise would be to write sin^5 x in terms of sin5x,sin4x,...

    • 2 years ago
  3. lgbasallote Group Title
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    ...i never really learned these stuffs...all i know are the basics of trigonometry....

    • 2 years ago
  4. lgbasallote Group Title
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    but i do have an idea how to do it......but it's brute force.....is there a shorter way?

    • 2 years ago
  5. hartnn Group Title
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    there are few ways i am aware of...idk what do u mean by brute force here....u can try to expand here,if u want.

    • 2 years ago
  6. lgbasallote Group Title
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    my idea is using sin (2x) over and over

    • 2 years ago
  7. joemath314159 Group Title
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    hmm...the first way that pops into my mind is using:\[e^{ix}=\cos(x)+i\sin(x)\]and being a little clever.

    • 2 years ago
  8. joemath314159 Group Title
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    Noting that sin(5x)= Im(cos(5x)+isin(5x))=Im(e^(5ix))=Im((e^ix)^5) =Im((cos(x)+isin(x))^5). So basically, if you see what the imaginary part of:\[(\cos(x)+i \sin(x))^5\]is, you will have a formula for sin(5x).

    • 2 years ago
  9. hartnn Group Title
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    nice! lets see how new learners try it... maybe they don't even know that e^{ix} identity

    • 2 years ago
  10. ParthKohli Group Title
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    I do know that formula made by Euler, but I didn't know we could apply that here. :P

    • 2 years ago
  11. hartnn Group Title
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    u can try it parth.... maybe @mathslover also wanna try this....

    • 2 years ago
  12. joemath314159 Group Title
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    Here is an example of using Euler's formula to show that:\[\cos(2x)=\cos^2(x)-\sin^2(x)\]\[\sin(2x)=2\sin(x)\cos(x)\]

    • 2 years ago
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  13. mathslover Group Title
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    Yeah , it is great to see the new pattern : @hartnn can you explain how to do it?

    • 2 years ago
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