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`;