trigonometric equations

trigonometric equations

A

trigonometric equation

values of given

trigonometric functions

whose

arguments

are unknown angles. The task is to determine all possible values of those angles. For obtaining the solution one needs the following

properties

of the trigonometric functions:

•

Two angles have the same value of sine iff the angles are equal or

supplementary angles

or differ of each other by a multiple of

full angle

.

•

Two angles have the same value of cosine iff the angles are equal or opposite (

http://planetmath.org/OppositeNumber

) angles or differ of each other by a multiple of full angle.

•

Two angles have the same value of

tangent

iff the angles are equal or differ of each other by a multiple of

straight angle

.

•

Two angles have the same value of

cotangent

iff the angles are equal or differ of each other by a multiple of straight angle.

The first principle in solving a trigonometric equation is that try to elaborate with

goniometric formulae

or else it so that only one trigonometric function on one angle remains in the equation. Then the equation is usually resolved to the form

f

⁢

(

k

⁢

x

)

=

a

,

(1)

where

k

and

a

are known numbers and

f

is one of the

functions

sin, cos, tan, cot. Thereafter one can solve the values of the angle

k

⁢

x

and, dividing these by

k

, at last the values of the angle

x

.

Example 1.

sin

⁡

x

⁢

cos

⁡

x

+

1

4

=

0

2

⁢

sin

⁡

x

⁢

cos

⁡

x

=

-

1

2

sin

⁡

2

⁢

x

=

-

1

2

sin

⁡

2

⁢

x

=

sin

⁡

210

∘

2

⁢

x

=

210

∘

+

n

⋅

360

∘

∨

2

⁢

x

=

180

∘

-

210

∘

+

n

⋅

360

∘

x

=

105

∘

+

n

⋅

180

∘

∨

x

=

-

15

∘

+

n

⋅

180

∘

On the third line one used the

double angle formula

of sine.

It may happen that the form (1) cannot be attained, but instead e.g. the form

f

⁢

(

k

⁢

x

)

=

f

⁢

(

α

)

,

(2)

where

x

can be in

α

.

Example 2.

sin

⁡

2

⁢

x

=

cos

⁡

3

⁢

x

cos

⁡

(

90

∘

-

2

⁢

x

)

=

cos

⁡

3

⁢

x

90

∘

-

2

⁢

x

=

3

⁢

x

+

n

⋅

360

∘

∨

90

∘

-

2

⁢

x

=

-

3

⁢

x

+

n

⋅

360

∘

-

2

⁢

x

-

3

⁢

x

=

-

90

∘

+

n

⋅

360

∘

∨

-

2

⁢

x

+

3

⁢

x

=

-

90

∘

+

n

⋅

360

∘

x

=

18

∘

+

n

⋅

72

∘

∨

x

=

-

90

∘

+

n

⋅

360

∘

On the second line one of the complement formulas was utilized.

Example 3.

sin

⁡

2

⁢

x

=

-

sin

⁡

3

⁢

x

sin

⁡

2

⁢

x

=

sin

⁡

(

-

3

⁢

x

)

2

⁢

x

=

-

3

⁢

x

+

n

⋅

360

∘

∨

2

⁢

x

=

180

∘

-

(

-

3

⁢

x

)

+

n

⋅

360

∘

x

=

n

⋅

72

∘

∨

x

=

180

∘

+

n

⋅

360

∘

On the second line the opposite angle formula (

http://planetmath.org/GoniometricFormulae

) of sine was utilized.

Example 4.

cos

⁡

2

⁢

x

=

-

cos

⁡

3

⁢

x

cos

⁡

2

⁢

x

=

cos

⁡

(

180

∘

-

3

⁢

x

)

2

⁢

x

=

±

(

180

∘

-

3

⁢

x

)

+

n

⋅

360

∘

x

=

36

∘

+

n

⋅

72

∘

∨

x

=

180

∘

+

n

⋅

360

∘

On the second line the supplement formula (

http://planetmath.org/GoniometricFormulae

) of sine was utilized.

Title

trigonometric equations

Canonical

name

TrigonometricEquations

Date of creation

2013-03-22 17:46:25

Last modified on

2013-03-22 17:46:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 33B10

Classification

msc 26A09

Classification

msc 00A35

Related topic

Equation

Defines

trigonometric equation

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