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Featured 5 months ago

Let us write the two complex numbers in polar coordinates and let them be

Here, if two complex numbers are

Their division leads us to

So for division complex number

Here

and

and

Hence,

Hence,

=

Featured 5 months ago

The Inverse Sine Function is denoted by,

is defined by,

The Inverse Sine Function is denoted by,

is defined by,

Featured 3 months ago

If

then

and

If

then

and

Plugging in the values for

If we call this value

then

and (since

Featured 2 months ago

Use the formula for a circle

The formula for a circle centred at the origin is

#x^2+y^2=r^2#

That is, the distance from the origin to any point

Picture a circle of radius

graph{(x^2+y^2-1)((x-sqrt(3)/2)^2+(y-0.5)^2-0.003)=0 [-2.5, 2.5, -1.25, 1.25]}

If we draw a line from that point to the origin, its length is

graph{(x^2+y^2-1)(y-sqrt(3)x/3)((y-0.25)^4/0.18+(x-sqrt(3)/2)^4/0.000001-0.02)(y^4/0.00001+(x-sqrt(3)/4)^4/2.7-0.01)=0 [-2.5, 2.5, -1.25, 1.25]}

Let the angle at the origin be theta (

Now for the trigonometry.

For an angle

#sin theta = "opp"/"hyp" = y/r" "<=>" "y=rsintheta#

Similarly,

#cos theta = "adj"/"hyp"=x/r" "<=>" "x = rcostheta#

So we have

#"Â Â Â Â Â Â "x^2"Â Â Â Â Â "+"Â Â Â Â Â Â "y^2"Â Â Â Â Â "=r^2#

#(rcostheta)^2+(rsintheta) ^2 = r^2#

#r^2cos^2theta + r^2 sin^2 theta = r^2#

The

#cos^2 theta + sin^2 theta = 1#

This is often rewritten with the

#sin^2 theta + cos^2 theta = 1#

And that's it. That's really all there is to it. Just as the distance between the origin and any point

Featured 1 week ago

Use the double angle formula for cosine to expand

Let

Featured yesterday

Taking

So

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