Trigonometry/Solving Trigonometric Equations

From Wikibooks, open books for an open world
Jump to navigation Jump to search

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.