Question

Solve the equation. tan²x + tan x - 2 = 0

Answer 1

Answer:

You can use a substitution, then solve the problem like a regular polynomial.

tan^2x-tanx=0

(tanx)^2-tanx=0

Now, let u=tanx:

u^2-u=0

u(u-1)=0

u=0,1

Plug tanx back in for u:

tanx=0,tanx=1

We know that tanx is sinx/cosx, so we can use that and solve each equation independently:

color(white){color(black)( (sinx/cosx=0,qquadsinx/cosx=1), (sinx=0,qquadsinx=cosx), (x=0", "pi", "2pi", "3pi..., qquad???):}

To figure out when sinx equals cosx, we can look at a unit circle:

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We can see that sinx=cosx when the angle is pi/4,(5pi)/4,(9pi)/4...

To write the general expression for 0,pi,2pi,3pi... we can use the letter k (or n, depending on the person) to represent "any integer".

This pattern would be x=pik, because sinx=0 when x is any multiple of pi.

The other pattern (pi/4, (5pi)/4, (9pi)/4...) can be rewritten as pi/4+0pi,pi/4+1pi,pi/4+2pi.... Now we can write a general expression for this using k: x=pi/4+pik

This means that the final solutions are:

x=pik, qquadpi/4+pik

Hope this helped!