1. An equation involving trigonometric functions of an unknown angle is called a trigonometric equation.
2. Basic trigonometric equations have the form
   \(\sin x = a,\) \(\cos x = a,\) \(\tan x = a,\) \(\cot x = a,\)
   where \(x\) is an unknown, \(a\) is any real number.

Equation \(\sin x = a\)

1. If \(\left| a \right| \gt 1\), the equation \(\sin x = a\) has no solutions.
2. If \(\left| a \right| \le 1,\) the general solution of the equation \(\sin x = a\) is written as
   \(x = {\left( { – 1} \right)^n}\arcsin a + \pi n,\) \(n \in \mathbb{Z}.\)
   This formula contains two branches of solutions:
   \({x_1} = \arcsin a + 2\pi n\),  \({x_2} = \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\).

1. In the simple case \(\sin x = 1\) the solution has the form
   \(x = \pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\).
2. Similarly, the solution of the equation \(\sin x = -1\) is given by
   \(x = -\pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\).
3. Case \(\sin x = 0\) (zeroes of the sine)
   \(x = \pi n,\) \(n \in \mathbb{Z}\).

Equation \(\cos x = a\)

1. If \(\left| a \right| \gt 1,\) the equation \(\cos x = a\) has no solutions.
2. If \(\left| a \right| \le 1,\) the general solution of the equation \(\cos x = a\) has the form
   \(x = \pm \arccos a + 2\pi n,\) \(n \in \mathbb{Z}.\)
   This formula includes two sets of solutions:
   \({x_1} = \arccos a + 2\pi n\), \({x_2} = -\arccos a + 2\pi n,\) \(n \in \mathbb{Z}\).

1. In the case \(\cos x = 1\), the solution is written as
   \(x = 2\pi n,\) \(n \in \mathbb{Z}\).
2. Case \(\cos x = -1\)
   \(x = \pi + 2\pi n,\) \(n \in \mathbb{Z}\).
3. Case \(\cos x = 0\) (zeroes of the cosine)
   \(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}\).

Equation \(\tan x = a\)

1. For any value of \(a\), the general solution of the equation \(\tan x = a\) has the form
   \(x = \arctan a + \pi n,\) \(n \in \mathbb{Z}.\)

1. Case \(\tan x = 0\) (zeroes of the tangent)
   \(x = \pi n,\) \(n \in \mathbb{Z}.\)

Equation \(\cot x = 0\)

1. For any value of\(a\), the general solution of the trigonometric equation \(\cot x = 0\) is written as
   \(x = \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}.\)

1. Case \(\cot x = 0\) (zeroes of the cotangent)
   \(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}.\)