Calculus
posted by Tracy .
Okay so I have a question on my assignment that says:
You are given that tan(y) = x. Find sin(y)^2. Express your answer in terms of x.
I know its derivatives, and I've tried taking the derivatives of both etc, and got them both to come out as cos(y)^2, which I know can be changed to 1sin(y)^2...but still that doesn't really do anything for me based on what the question asks right? I'm so lost, help would be appreciated. Thanks
I suppose you could just solve this directly as tan(y)=x, then arctan(x)=y. So sin(artan(x))=sin(y) and
sin^{2}(artan(x))=sin^{2}(y)
But it semms you're expected to solve this a specific way.
There's also a trig identity that says
If arctan(z) = u/2 then sin(u) = 2z/(1+z^{2})
In terms of your vaiables, tan(y)=x or arctan(x)=y and
sin(2y) = 2x/(1+x^{2}) or
If you're using derivatives then
d/dy tan(y) = sec^{2}(y), or
dx/dy = sec^{2}(y) and
dy/dx = cos^{2}(y) = 1sin^{2}(y) so
sin^{2}(y) = 1  dy/dx
If x=f(y)=tan(y) then dy/dx = d/dy f^{1} = d/dx arctan(x) = 1/(1+x^{2})
So sin^{2}(y) = 1  1/(1+x^{2}) =
x^{2}/(1+x^{2})
Be sure to check my work and verify the result for yourself too.
There are a number of ways to solve this, so I suppose the answer depends on what you're studying right now.
The question does not look like calculus to me. It looks like trig and algebra. No derivatives necessary.
tan y = sin y/cos y = sqrt[sin^2 y/(1  sin^2 y)]
tan^2 y = x^2 = [sin^2 y/(1  sin^2 y)]
x^2 (1  sin^2 y) = sin^2 y
sin^2 y (1 + x^2)= x^2
sin^2 y = x^2/(1 + x^2)
Check the algebra. Sometimes I get slopy
Nope,the algebra's just fine. That's probably the clearest explanation here.
You're right, this is just plain ol' trig manipulation and I really don't see how derivatives fit in, unless they wanted something to do with the inverse function,...I didn't see it either.
Thanks guys. I wasn't really seeing how the derivcatives fit in either which is why I was getting confused.
You're welcome. Let me just add that derivatives 'could' fit in, but drwis's solution is without a doubt the simplest way to approach this particular problem. If you take a look at the first 'solution' I proposed, it would work if you simply needed to calculate something say once. But if you were writing a computer program then you'd definitely want to find an identity with the least calculation overhead; using sin^2 y = x^2/(1+x^2) would be what to use in a program and drwis's sol'n would be the way to find it.
Ohhh okay I get it. Thanks soo much = )
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