# 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

### 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