How do we solve linear trigonometric equations?

Aim: How do we solve linear trigonometric equations? 

Solving trigonometric equations are easy as long as you know how to do them.  

There are three steps to solving linear equations:
1)  Get the trig function alone.
2)  Use inverse to solve for angle or use the helpful Unit Circle!
3)  Check for other solutions.

Example: 

Solve for all values between 0 to 2:
2 sin θ-√3 = 0
1) To get the trig function alone, add √3 to both sides of the equation.
                             2 sinθ -√3 = 0
                                       +√3   +√3
                             2 sinθ = √3
2) Divide 2 from both sides to get sinθ completely alone.
                              
                                 2 sinθ = √3
                                    2          2
                                 sinθ =  √3
                                              2
3) Now, if you look at the unit circle, you will see that the only places where sin (y-coordinate) is √3 are at π and 2π or in degrees 60° 
                                               2            3         3
and 120°. These are the solutions. You can also achieve these solutions by first using the calculator to solve for inverse sine of √3, 
                                                                                                       2
which will give you 60° and you can subtract it from 180 to get 120° since sine is only positive in the first two quadrants or 180°
Example #2:

tan θ -­ 2 cos θ tan θ = 0

1) Factor to solve for both tanθ and cosθ. Factor out tanθ since it is 

common. 

                                  tan θ -­ 2 cos θ tan θ = 0
                                  
                                   tanθ (1- 2 cosθ) = 0

2) Set each equal to 0 and solve.


                    tanθ = 0               and              1-2 cosθ = 0

                                                                     -1           -1


                                                                    -2 cosθ = -1

                                                                     -2            -2
                                                                      
                                                                      cosθ = 1/2




3) If you look at the unit circle, you will see that tangent (sin/cos) 

equals to 0 at 0° and 180° or 0 and π in radians. The cosine(x-

coordinate) are 60° and 300° or π/3 and 5π/3. These are the 

solutions.