Solve each of the following. Please give both the general solution and the specific solutions in the given interval.

1. 2sin(2theta)+sqrt3=0 interval [0,2pi)

2. 2sin^2x=2+cosx interval [0,pi]

3. tan(3x)(tan x-1)=0 interval [0, 2pi)

Please work through it with me I want to learn! !

Question

Solve each of the following. Please give both the general solution and the specific solutions in the given interval.

1. 2sin(2theta)+sqrt3=0 interval [0,2pi)

2. 2sin^2x=2+cosx   interval [0,pi]

3. tan(3x)(tan x-1)=0 interval [0, 2pi)

Please work through it with me I want to learn! !

e.mccormick · Answer

Know how you take a polynomial and factor it and then solve each factor?

$x^2-x-2=0\implies (x+1)(x-2)=0\implies $
$x+1=0$ and $x-2=0$ $\therefore x=\{-1,2\}$

Well, this will be the same sort of thing.  You get it to a factored form, then solve.

e.mccormick · Answer

Now, where it gets different is when you are dealing with the number of answers.  There can be multiple angles that will have the same answer!  This is best shown using the unit circle:
|dw:1381802369960:dw|
In these three angles, two share an x, and two share a y.  Well, that means that the cosine of the shared x is the same, but the angles are different!  Similarly, the sine of the hared y is the same, and the angles are again different.

If I went around the circle again and again and again, I could have infinitely many angles on these three spots!

e.mccormick · Answer

A good reference or three on this: http://www.regentsprep.org/regents/math/algtrig/ATT10/trigequations.htm http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx http://www.purplemath.com/modules/solvtrig.htm I hope all that helps!