# The Simplest Trigonometric Equations

Equation sin(x)=a where |a|<=1 has infinitely many roots. For example roots of the equation sin(x)=1/2 are following: x_1=(pi)/6,x_2=(5pi)/6,x_3=pi/6+2pi,x_4=pi/6-2pi etc. The common formula, that allows to find all roots of the equation sin(x)=a is following:

Fact 1. If sin(x)=a, where |a|<=1 then x=(-1)^n arcsin(a)+pin,n in Z.

Here n can take integer values, and for any value of n we obtain some root of the equation.

Fact 2. If cos(x)=a, where |a|<=1 then x=+- arccos(a)+2pin,n in Z.

Fact 3. If tan(x)=a, then x=arctan(a)+pin,n in Z.

Fact 4. If cot(x)=a, then x=text(arccot)(a)+pin,n in Z.

Example 1. Solve sin(x)=1/2.

Using Fact 1 we obtain that x=(-1)^n arcsin(1/2)+pin,n in Z. Since arcsin(1/2)=pi/6, then finally we obtain that x=(-1)^n pi/6+pin, n in Z.

Example 2. Solve cos(3x)=-1/(sqrt(2)).

Using Fact 2 we obtain that 3x=+-arccos(-1/(sqrt(2)))+2pin,n in Z. Since arccos(-1/(sqrt(2)))=pi-arccos(1/(sqrt(2)))=pi-pi/4=(3pi)/4, then we obtain that 3x=+- (3pi)/4+2pin, n in Z or x=+-(pi)/4+(2pin)/3, n in Z.

Example 3. Solve tan(x-pi/6)=-sqrt(3).

According to Fact 3 x-pi/6=arctan(-sqrt(3))+pin,n in Z. Since arctan(-sqrt(3))=-arctan(sqrt(3))=-pi/3, then we obtain that x-pi/6=-pi/3+pin, n in Z, i.e. x=-pi/6+pin, n in Z.

Note, that in some cases it is more convenient to use particular formulas (in all formulas n in Z):

1. If sin(x)=0 then x=pin;
2. If sin(x)=1 then x=pi/2+2pin;
3. If sin(x)=-1 then x=-pi/2+2pin;
4. If cos(x)=0 then x=pi/2+pin;
5. If cos(x)=1 then x=2pin;
6. If cos(x)=-1 then x=pi+2pin;
7. If tan(x)=0 then x=pin;
8. If cot(x)=0 then x=pi/2+pin;