# 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: \(\arcsin a,\) \(\arccos a,\) \(\arctan a,\) \(\text {arccot }a\)

An equation involving trigonometric functions of an unknown angle is called a trigonometric equation.

Basic trigonometric equations have the form

here \(x\) 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

This formula contains two branches of solutions:

The solutions of a trigonometric equation that lie in the interval \(\left[ {0,2\pi } \right)\) are called principal solutions. The principal solutions for the equation \(\sin x = a\) are

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

Similarly, the solution of the equation \(\sin x = -1\) is given by

Case \(\sin x = 0\) (zeroes of the sine):

## 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

This formula includes two sets of solutions:

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

Case \(\cos x = -1:\)

Case \(\cos x = 0\) (zeroes of the cosine):

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

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

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

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

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

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

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Solve the equation

### Example 2

Solve the equation

### Example 3

Find the general solution of the equation

### Example 4

Find the principal solutions of the equation

### Example 5

Solve the equation

### Example 6

Solve the equation

### Example 1.

Solve the equation

Solution.

The general solution of the equation has the form

The inverse sine function is odd, so that

Hence, the general solution is written as

The principal solutions on the interval \(\left[ {0,2\pi } \right)\) are given by

### Example 2.

Solve the equation

Solution.

In this special case, the general solution is given by

Solve it for \(x:\)

The principal solution contains one value:

### Example 3.

Find the general solution of the equation

Solution.

We rewrite the equation in the form

and divide both sides by \(\cos x.\) Note that \(\cos x \ne 0.\) Indeed, if \(\cos x = 0,\) then

that is, \(\cos x = 0\) cannot be a solution of the equation. So we have

The general solution is given by

### Example 4.

Find the principal solutions of the equation

Solution.

First we find the general solution. Given that \(\text{arccot}\left( { - a} \right) = \pi - \text{arccot } a,\) we have

where \(n \in \mathbb{Z}.\)

The principal values lie in the interval \(\left[ {0,2\pi } \right).\) Hence, our principal solutions will be

### Example 5.

Solve the equation

Solution.

This equation has two solutions:

Substitute the values of inverse cosine:

So the general solution is given by

where \(n, k \in \mathbb{Z}.\)

Respectively, the principal solutions of the equation are

### Example 6.

Solve the equation

Solution.

We rewrite the equation as follows:

We have obtained two equations. The first equation \(\tan x = 1\) has the following solution:

The second equation has a solution in the form

We can merge both solutions and express them with one formula:

The principal values are given by