Trig?!
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sin6x = sin (4x + 2x) = sin4x*cox2x + cos4x*sin2x
sin2x = sin (4x - 2x) = sin4x*cox2x - cos4x*sin2x
So,
sin6x + sin2x = 2sin4x*cox2x
And,
sin2x + 2sin4x + sin6x
= sin6x + sin2x + 2sin4x
= 2sin4x*cox2x + 2sin4x
= 2sin4x (cox2x + 1)
= 4cos^2x•sin4x
QED.
sin2x = sin (4x - 2x) = sin4x*cox2x - cos4x*sin2x
So,
sin6x + sin2x = 2sin4x*cox2x
And,
sin2x + 2sin4x + sin6x
= sin6x + sin2x + 2sin4x
= 2sin4x*cox2x + 2sin4x
= 2sin4x (cox2x + 1)
= 4cos^2x•sin4x
QED.
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In order to do this, you MUST learn your trig identities:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) - sin²(x)
= 2cos²(x) - 1
= 1 - 2sin²(x)
tan(2x) = 2tan(x) / [1 - tan²(x)]
It is absolutely VITAL that you learn these identities and how to manipulate them if you want to be successful at A level (I think you are doing C3 or C4) Maths and any further maths (e.g A Level further maths, uni, etc) that you wish to do.
Now, before we answer your question, you must learn where the identities come from. I will give you the sine one and then you must learn the rest yourself:
By definition sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
Therefore, we can see that sin(2x) = sin(x + x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x).
The reason I have told you this is because you might forget the identity for sin(4x) and sin(6x) and so you would have to work it out, and therefore you must know the formula.
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) - sin²(x)
= 2cos²(x) - 1
= 1 - 2sin²(x)
tan(2x) = 2tan(x) / [1 - tan²(x)]
It is absolutely VITAL that you learn these identities and how to manipulate them if you want to be successful at A level (I think you are doing C3 or C4) Maths and any further maths (e.g A Level further maths, uni, etc) that you wish to do.
Now, before we answer your question, you must learn where the identities come from. I will give you the sine one and then you must learn the rest yourself:
By definition sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
Therefore, we can see that sin(2x) = sin(x + x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x).
The reason I have told you this is because you might forget the identity for sin(4x) and sin(6x) and so you would have to work it out, and therefore you must know the formula.