Solve Trig Question: tan(2θ) = cot(φ)

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SUMMARY

The equation tan(2θ) = cot(φ) can be simplified using trigonometric identities. The correct approach is to recognize that cot(φ) is equivalent to tan(π/2 - φ), leading to the conclusion that 2θ = π/2 - φ. Alternatively, using the identity tan(2θ) = 2tan(θ)/(1 - tan²(θ)), one can derive the same relationship. Both methods yield the same result, but the first method is more straightforward.

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[SOLVED] Trig Question

Homework Statement


Let's say I have [tex]tan(2\theta) = cot(\phi)[/tex] I'm not quite sure how to simplify it (I want to solve for theta) I could say [tex]tan(2\theta) = 1 / tan(\phi)[/tex]
[tex]2\theta = atan( 1 / tan(\phi) )[/tex] but then I'm not sure what to do after that.
Do I take the atan of 1 and [tex]tan(\phi)[/tex]?

I know another method to find it is to say [tex]cot(\phi) = \pi/2 - tan(\phi)[/tex] but I don't understand that either because [tex]2\theta = atan( \pi/2 - tan\phi ) != \pi/2 -\phi[/tex] (not exactly at least)
The answer as far as I know is supposed to be [tex]2\theta = \pi/2 -\phi[/tex]
 
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Remember that that the tangent of angle is the cotangent of its complement and go from that definition.

OR you could say that


[tex]tan(2\theta) = cot(\phi) \Rightarrow \frac{2tan\theta}{1-tan^2\theta}=\frac{1}{tan\phi}[/tex]

and solve from there.

Two ways really, one is easier than the other though.
 

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