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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:
$${x_1} = \arcsin a + 2\pi n$$,  $${x_2} = \pi – \arcsin a + 2\pi n,$$ $$n \in \mathbb{Z}$$.
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)
$$x = \pi/2 + \pi n,$$ $$n \in \mathbb{Z}.$$