Trigonometric equations

Elementary trigonometric equations

` sin(x) = a `

The equation has solutions if

Then,

Special cases

` cos(x) = a `

The equation has solutions if

Then,

Special cases

Solution methods

Variable substitution

Trigonometric equations can be reduced to algebraic equations through variable substitution. For example, the equation is reduced to a plain quadratic equation by rearranging and substituing

Example

Solve trigonometric equation:

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Perform substitution: , where

Substitute back:

Example

Solve equation:

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Domain:

Note that is already covered by the domain of

Then, substituting back:

Reducing to ` f(x)*g(x) = 0 `

If possible to rewrite as , then it follows that and/or . Both equations are then solved separately.

Example

Solve:

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Domain: because of

Then,

and

…which is not in the set of permissible values.

Finally,

Reducing to ` f(x)/g(x) = 0 `

If possible to rewrite as , then, given that , the whole equation is reduced to by multiplying both sides by

Example

Solve:

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Note that:

Then,

Define the set of permissible values:

Then, we need to solve

Solving :

Since the last solution is not in the set of permissible values,

Homogeneous trigonometric equations

Homogeneous trigonometric equations are defined as homogeneous polynomials of equal to 0:

… or, if expanded:

where

These equations are solved by dividing by or

Example

Solve:

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Equations reducible to homogeneous

Equations of the form:

can sometimes be reduced to normal homogeneous equations by rewriting as . Given a passing form of , subtracting from both sides will yield a normal homogeneous equation.

Example

Solve:

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could be a homogeneous equation of degree 3. This can be indeed achieved by rewriting as:

…which is a homogeneous equation, solved in the usual way.

… and so on.

Form ` R(sin(nx), cos(kx), tan(mx), cot(lx)) = 0 `

Equations of form where is a rational function are solvable through universal trigonometric substitution . Then, using these trig formulas, it follows that:

Example

Solve:

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The equation conforms to because it can be rewritten as

In this case, we can think of the denominator as simply or similar.

Perform substitution: . Then,

Note that

These equations are elementary and solved correspondingly.

Homogeneous equations of 1st degree

Equations of shape

where are constants.

Methods: universal trigonometric substitution or the method of auxilliary angle.

Method of auxilliary angle: divide both sides by . Then:

for some angles in a right triangle. Then solve using identities of trigonometric functions of sum of arguments

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Assume a right triangle whose legs are equal to from the equation. Then the hypotenuse of this triangle is . Then, when we divide both sides of the equation by , we receive:

If we arbitrarily choose an acute angle (resp. ) in this right triangle, is an adjacent (resp. opposite) side to it, with (resp. ), and is opposite (resp. adjacent) side to it, with (resp. ).

Then we can write this equation as

which can be then solved using trigonometric identities to rewrite the left side as a trigonometric function of sum of arguments.

Example

Solve:

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Start by dividing both sides by the hypotenuse of a right triangle with legs :

It does not matter if we assign or for some angle , see both solutions:

` sqrt(3)/2 = cos(alpha) `
` sqrt(3)/2 = sin(alpha) `

Shape `R(sinx +- cosx; sinx*cosx)`

Method: substitution

Then:

Example

Solve:

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The latter equations are solved using one of the methods described above.