# Formulas and Tables

Trigonometry# Basic Trigonometric Equations

Set of integers: \(\mathbb{Z}\)

Integer: \(n\)

Real number: \(a\)

Inverse trigonometric functions: \(\arcsin a,\) \(\arccos a,\) \(\arctan a,\) \(\text {arccot }a\)

- An equation involving trigonometric functions of an unknown angle is called a trigonometric equation.
- 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 \(\sin x = a\)

- If \(\left| a \right| \gt 1\), the equation \(\sin x = a\) has no solutions.
- 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:

\({x_1} = \arcsin a + 2\pi n\), \({x_2} = \pi – \arcsin a + 2\pi n,\) \(n \in \mathbb{Z}\).

- In the simple case \(\sin x = 1\) the solution has the form

\(x = \pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\). - Similarly, the solution of the equation \(\sin x = -1\) is given by

\(x = -\pi/2 + 2\pi n,\) \(n \in \mathbb{Z}\). - Case \(\sin x = 0\) (zeroes of the sine)

\(x = \pi n,\) \(n \in \mathbb{Z}\).

### Equation \(\cos x = a\)

- If \(\left| a \right| \gt 1,\) the equation \(\cos x = a\) has no solutions.
- 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:

\({x_1} = \arccos a + 2\pi n\), \({x_2} = -\arccos a + 2\pi n,\) \(n \in \mathbb{Z}\).

- In the case \(\cos x = 1\), the solution is written as

\(x = 2\pi n,\) \(n \in \mathbb{Z}\). - Case \(\cos x = -1\)

\(x = \pi + 2\pi n,\) \(n \in \mathbb{Z}\). - Case \(\cos x = 0\) (zeroes of the cosine)

\(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}\).

### Equation \(\tan x = a\)

- For any value of \(a\), the general solution of the equation \(\tan x = a\) has the form

\(x = \arctan a + \pi n,\) \(n \in \mathbb{Z}.\)

- Case \(\tan x = 0\) (zeroes of the tangent)

\(x = \pi n,\) \(n \in \mathbb{Z}.\)

### Equation \(\cot x = 0\)

- For any value of\(a\), the general solution of the trigonometric equation \(\cot x = 0\) is written as

\(x = \text {arccot } a + \pi n,\) \(n \in \mathbb{Z}.\)

- Case \(\cot x = 0\) (zeroes of the cotangent)

\(x = \pi/2 + \pi n,\) \(n \in \mathbb{Z}.\)

### Related Pages

- Definition and Graphs of Trigonometric Functions
- Basic Trigonometric Identities
- Cofunction and Reduction Identities
- Relationships between Trigonometric Functions
- Addition and Subtraction Formulas
- Double and Multiple Angle Formulas
- Half-Angle Formulas
- Sum-to-Product Identities
- Product-to-Sum Identities
- Derivatives of Trigonometric Functions
- The Indefinite Integral and Basic Formulas of Integration. Table of Integrals.
- Basic Trigonometric Inequalities
- Equations