# Homogeneous Trigonometric Equations

A trigonometric equation is said to be a homogeneous equation in \(\sin x\) and \(\cos x\) if the degrees of all terms in the equation are the same. This degree is called the degree of the homogeneous equation.

For example, the equation of the form

is a first-order homogeneous equation. Notice that a homogeneous equation has zero on the right-hand side of the equality sign.

A second-order homogeneous trigonometric equation is written as

A similar second-order equation of the form

is not a homogeneous equation since its right-hand side is not equal to zero. But we can reduce it to a homogeneous equation using the Pythagorean identity:

A third-order homogeneous trigonometric equation is given by

and so on.

By dividing by \({\cos ^n}x,\) a homogeneous equation of degree \(n\) can be reduced to an algebraic equation with respect to the function \(\tan x.\)

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Solve the equation

### Example 2

Find the general solution of the equation

### Example 3

Solve the trigonometric equation

### Example 4

Solve the equation

### Example 5

Solve the trigonometric equation

### Example 6

Find the general solution of the equation

### Example 1.

Solve the equation

Solution.

We divide both sides of the equation by \(\cos x.\) This can be done because \(\cos x \ne 0.\) Indeed, cosine and sine in the left-hand side cannot be simultaneously equal to zero, so \(\cos x = 0\) is not a solution of the equation.

The basic equation \(\tan x = -1\) has the following solution:

### Example 2.

Find the general solution of the equation

Solution.

We deal here with a second-order homogeneous equation. Divide both sides by \({\cos ^2}3x\) and take the square root:

Determine the general solution to this equation:

### Example 3.

Solve the trigonometric equation

Solution.

By the double-angle formula for sine, we can write

Divide by \({\cos ^2}x:\)

By denoting \(\tan x = t,\) determine the roots of the quadratic equation:

We get two families of solutions:

Answer:

### Example 4.

Solve the equation

Solution.

We have here a linear nonhomogeneous equation. However, using the double-angle formulas and Pythagorean identity, we can convert it into a second-order homogeneous equation:

or

Divide both sides by \({\cos ^2}x:\)

We got a quadratic trigonometric equation. Making the substitution \(\tan x = t,\) find the roots of the equation:

Hence, the original equation has two sets of solutions:

### Example 5.

Solve the trigonometric equation

Solution.

This is not a homogeneous equation since not all terms have the same degree. To make it homogeneous, we multiply the last two terms in the left-hand side by \(1 = {\sin ^2}x + {\cos ^2}x.\) This yields:

Now we have a fourth-degree homogeneous equation and can solve it by dividing by \({\cos ^4}x:\)

Let \({\tan ^2}x = t.\) Then

The first root \({t_1} = - \frac{1}{2}\) does not produce any solutions:

The second root \({t_2} = \frac{1}{3}\) gives us the following solution:

### Example 6.

Find the general solution of the equation

Solution.

Note that the domain of the equation is defined by the condition \(\cos x \ne 0.\) Let's rewrite the right-hand side of the equation:

Now we have a second-order homogeneous equation:

Factor the left-hand side:

The first set of solutions is given by

The other set of solutions follows from the first-order homogeneous equation:

Hence, the general solution is given by