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.

Contents

Basic trigonometric equationsEdit

sin(x) = nEdit

 
   
   
   
   
   
   

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) = nEdit

 
   
   
   
   
   
   

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) = nEdit

 
   
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) = nEdit

 
   
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) = nEdit

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

 
 

Further examplesEdit

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) = cEdit

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.