Question 1173665
1) To find a general solution, first use algebra to transform the equation into the form:
trigfunction(something) = number<br>
3 = 2sin (3(x+ (pi/3)) +4<br>

Subtracting 4 from each side we get:
-1 = 2sin (3(x+ (pi/3))<br>

Dividing both sides by 2 we get:
-1/2 = sin (3(x+ (pi/3))
which is the desired form.<br>

2) Next, we need to find expressions for angles whose sin's are -1/2. Since sin(pi/6) = 1/2 the reference angle is pi/6. And since sin is negative in the 3rd and 4th quadrants, we are looking for angles in those quadrants with a  reference of pi/6.<br> 

In the third quadrant our expression would be:
pi + pi/6 + 2pi*n
which simplifies to:
7pi/6 + 2pi*n<br>

In the fourth quadrant our expression would be:
2pi - pi/6 + 2pi*n
which simplifies to:
11pi/6 + 2pi*n<br>

NOTE: The 2pi*n is a way for us to include all the coterminal angles (which would also all have sin values of -1/2. The "n" can be replace by any integer. Each different integer results in another angle whose sin is -1/2.<br>

3) Write equations to set the trig function's argument to the expressions from step 2:<br>

3(x+ (pi/3)) = 7pi/6 + 2pi*n
which simplifies to
3x + pi = 7pi/6 + 2pi*n<br>

3(x+ (pi/3)) = 11pi/6 + 2pi*n
which simplifies to
3x + pi = 11pi/6 + 2pi*n<br>

4) Solve these equations for x.
Subtracting pi from each side of both equations:
3x = pi/6 + 2pi*n
3x = 5pi/6 + 2pi*n<br>
Multiplying each side by 1/3 (or dividing both sides by 3) we get:
x = pi/18 + (2/3)pi*n
x = 5pi/18 + (2/3)pi*n<br>

These last equations are the "general solution".