Trigonometry/Solving Trigonometric Equations

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Trigonometric equations are equations including trigonometric functions. If they have only such functions and constants, then the solution involves finding an unknown which is an argument to a trigonometric function.

Basic trigonometric equations[edit]

sin(x) = n[edit]

Sin unit circle.svg

The equation has solutions only when is within the interval . If is within this interval, then we first find an such that:

The solutions are then:

Where is an integer.

In the cases when equals 1, 0 or -1 these solutions have simpler forms which are summarized in the table on the right.

For example, to solve:

First find  :

Then substitute in the formulae above:

Solving these linear equations for gives the final answer:

Where is an integer.

cos(x) = n[edit]

Cos unit circle.svg

Like the sine equation, an equation of the form only has solutions when n is in the interval . To solve such an equation we first find one angle such that:

Then the solutions for are:

Where is an integer.

Simpler cases with equal to 1, 0 or -1 are summarized in the table on the right.

tan(x) = n[edit]

Tan unit circle.svg
General
case

An equation of the form has solutions for any real . To find them we must first find an angle such that:

After finding , the solutions for are:

When equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.

cot(x) = n[edit]

Cot unit circle.svg
General
case

The equation has solutions for any real . To find them we must first find an angle such that:

After finding , the solutions for are:

When equals 1, 0 or -1 the solutions have simpler forms which are shown in the table on the right.

csc(x) = n and sec(x) = n[edit]

The trigonometric equations and can be solved by transforming them to other basic equations:

Further examples[edit]

Generally, to solve trigonometric equations we must first transform them to a basic trigonometric equation using the trigonometric identities. This sections lists some common examples.

a sin(x)+b cos(x) = c[edit]

To solve this equation we will use the identity:

The equation becomes:

This equation is of the form and can be solved with the formulae given above.

For example we will solve:

In this case we have:

Apply the identity:

So using the formulae for the solutions to the equation are:

Where is an integer.