Finding all values that satisfy the equation

[EL ALEM]

Homework Statement

Find all values of x in the interval [0,2pi] that satisfy:

tan^2(x)-1=(2/sqrt3)tan(x)

Homework Equations

Quadratic formula

The Attempt at a Solution

Is there anyway to this without using the quadratic formula that will get me exact answers?
Also will using the quadratic formula give me exact answers?

 

[Mentallic]

Not that I know of, but if there is (there most likely is) then it would involve trig substitutions and such and would involve much more work and complications than just using the quadratic formula.

Yes, of course it will! Do you trust the quadratic formula when the quadratic is in terms of x? Well then it works for quadratics in terms of u where u=f(x), in this case, u=tan(x).

 

[The legend]

i think the quadratic formula would be the easiest...
And the exact answers as you say can be found only for a particular interval [-pi/2,pi/2]...or else you get a general solution...

 

[EL ALEM]

Yup thanks for the reply, I ended up doing using the quadratic formula and ended up simplifying nicely (i got tanx= -sqrt3/3, sqrt3)

but one question how would i find ALL the values on [o,2pi], i know how to find the ones in the first quadrant (i.e. tanx= sqrt3 => x=pi/3 ) but how do i find the rest? I forget how to do that.

 

[The legend]

you know general soln of tan(x)=y is
x = tan-1(y) + k*pi , where k can be any integer.
just substitute different values of k and get your answer...

 

[EL ALEM]

Ok is it the same thing for sin and cos? thanks in advance.

 

[The legend]

no it isnt...
http://mathsfirst.massey.ac.nz/Trig/TrigGenSol.htm

 

[The legend]

did you understand it?

 

[EL ALEM]

Yup that cleared A LOT of things up. Thanks a mill.