Precalculus

Trigonometry

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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:

\[{x_1} = \arcsin a + 2\pi n,\;{x_2} = \pi - \arcsin a + 2\pi n,\;n \in \mathbb{Z}.\]
Solution of the equation sin(x)=a
Figure 1.

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:

\[{x_1} = \arccos a + 2\pi n,\; {x_2} = -\arccos a + 2\pi n,\; n \in \mathbb{Z}.\]
Solution of the equation cos(x)=a
Figure 2.

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

\[x = \arctan a + \pi n,\; n \in \mathbb{Z}.\]
Solution of the equation tan(x)=a
Figure 3.

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

\[x = \text {arccot } a + \pi n,\; n \in \mathbb{Z}.\]
Solution of the equation cot(x)=a
Figure 4.

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

\[x = \pi/2 + \pi n,\; n \in \mathbb{Z}.\]