# Precalculus

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

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

Basic trigonometric equations have the form

$\sin x = a,\;\cos x = a,\;\tan x = a,\;\cot x = a,$

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

$x = {\left( { - 1} \right)^n}\arcsin a + \pi n,\;n \in \mathbb{Z}.$

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