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how is x=tanθ when using trig sub

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For (a^2+x^2), you substitute x = a tan (theta) because when you replace the equation a^2 + x^2 with a^2 + (a tan (theta))^2, you get a^2 ( 1 + tan^2 (theta)) which becomes a^2 (sec^2 (theta)).
I'm trying to figure out when solving for arc length r=e^θ from (1,0) to origin, you have to convert the limits in terms of theta. I found that someone used tanθ = x to solve the problem. I'm wondering how the person got the equation.
For arc length, can you use the equation \[\int\limits_(\sqrt(1+(f'(x))^2) \] from a to b?

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∫ab sqrt (r^2 + (dr/dθ)^2) dθ is what i used
Are you doing it in polar?
You are integrating dθ - so you need the limits in the integral to be in terms of theta. i'm confused about the trig sub that he used to convert the limits
Well you'd still use a a tan \[\theta\] because it will simplify if you use trig identities.
\[a \tan(\theta)\]
but how is tanθ = x or y
isnt it tanθ = y/x?

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