Solve `tan(x)+sqrt(3)=0 ` :
The tangent has value `sqrt(3) ` when the sine has value `sqrt(3)/2 ` and the cosine has value `1/2 ` . (`tanx=sinx/cosx ` )
The tangent is positive in the first and third quadrants, and negative in the second and fourth quadrants.
Thus `x=-pi/3+npi ` for ` ``n in ZZ ` (n an integer.)
The graph of tan(x) and the line `y=-sqrt(3) ` :
Fist simplify the equation, like so:
tan x = -3^(1/2)
tan x as a rule is equal to sinx / cosx. From there you work backwards to determine what values of sin and cos would equal to negative square root of 3. If the results of sin were square root of 3 /2 and cos was 1/2, then the 2s will cancel, leaving you with square root of 3. Now to determine which angle on the unit circle will give you a negative value. This can happen in the 2nd or 4th quadrant at x = 120 degrees or 300 degrees, or 2pi/3 and 5pi/3 in radians.
``But we know
where n is an integer.
You will need to memorize the unit circle to solve similar problems. The +n(pi) part of the solution arises from the fact that trigonometric functions repeat themselves in periodic cycles.