Formulas and Tables

Trigonometry

Basic Trigonometric Equations

Angles (arguments of functions): \(x,\) \({x_1},\) \({x_2}\)
Set of integers: \(\mathbb{Z}\)
Integer: \(n\)
Real number: \(a\)

Trigonometric functions: \(\sin x,\) \(\cos x,\) \(\tan x,\) \(\cot x\)
Inverse trigonometric functions: \(\arcsin a,\) \(\arccos a,\) \(\arctan a,\) \(\text {arccot }a\)

  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:
    \({x_1} = \arcsin a + 2\pi n\),  \({x_2} = \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\).
Solution of the equation sin(x)=a
  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:
    \({x_1} = \arccos a + 2\pi n\), \({x_2} = -\arccos a + 2\pi n,\) \(n \in \mathbb{Z}\).
Solution of the equation cos(x)=a
  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
    \(x = \arctan a + \pi n,\) \(n \in \mathbb{Z}.\)
Solution of the equation tan(x)=a
  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
    \(x = \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}.\)
Solution of the equation cot(x)=a
  1. Case \(\cot x = 0\) (zeroes of the cotangent)
    \(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}.\)