Trigonometric equations. Main methods for solving

Trigonometric equations. Simplest trigonometric equations. 
Methods of solving:  algebraic method,  factoring, reducing 
to a homogeneous equation, transition to a half-angle, 
introducing an auxiliary angle, transforming a product to 
a sum, universal substitution.

Trigonometric equations. An equation, containing an unknown under the trigonometric function sign is called trigonometric.

Simplest trigonometric equations.

Methods for solving trigonometric equations. Solving of a trigonometric equation consists of the two stages: transforming of a equation to receive its simplest shape ( see above ) and solving of the received simplest trigonometric equation. There are seven main methods of solution of trigonometric equations.

1. Algebraic method. This method is well known for us from algebra ( exchange and
    substitution method ).

2. Factoring.  Consider this method by examples.

    E x a m p l e  1.  Solve the equation:  sin x + cos x = 1 .

    S o l u t i o n .    Transfer all terms to the left:

sin x + cos x – 1 = 0 ,

                               transform and factor the left-hand side expression:

    E x a m p l e  2.  Solve the equation:  cos² x + sin x · cos x = 1.

    S o l u t i o n .     cos ² x + sin x · cos x – sin ² x – cos ² x = 0 ,

sin x · cos x – sin ² x = 0 ,

sin x · ( cos x – sin x ) = 0 ,

    E x a m p l e  3.  Solve the equation:  cos 2x – cos 8x + cos 6x = 1.

    S o l u t i o n .     cos 2x + cos 6x = 1 + cos 8x ,

                              2 cos 4x cos 2x = 2 cos ² 4x ,

                               cos 4x · ( cos 2x –  cos 4x ) = 0 ,

                              cos 4x · 2 sin 3x · sin x = 0 ,

                              1).  cos 4x = 0 ,               2).  sin 3x = 0 ,          3). sin x = 0 ,

4. Transition to a half-angle. Consider this method by the example:

    E x a m p l e .  Solve the equation:  3 sin x – 5 cos x = 7.

    S o l u t i o n .  6 sin ( x / 2 ) · cos ( x / 2 ) – 5 cos ² ( x / 2 ) + 5 sin ² ( x / 2 ) =

                                                                       = 7 sin ² ( x / 2 ) + 7 cos ² ( x / 2 ) ,

                            2 sin ² ( x / 2 ) – 6 sin ( x / 2 ) · cos ( x / 2 ) + 12 cos ² ( x / 2 ) = 0 ,

                            tan ² ( x / 2 ) – 3 tan ( x / 2 ) + 6 = 0 ,

                                     .   .   .   .   .   .   .   .   .   .

5. Introducing an auxiliary angle. Consider an equation of the shape:

                                           a sin x + b cos x = c ,

    where  a, b, c – coefficients;  x – an unknown.

    New coefficients in the left-hand side have the properties of sine and cosine: a modulus 
    ( absolute value ) of each of them is not more than 1, and a sum of their squares is equal 
    to 1. So, we can mark them as  cos φ and sin φ correspondingly; here   is the so called 
    an auxiliary angle.

    Now our equation has the shape:

6. Transforming a product to a sum. The corresponding formulas are used here.

    E x a m p l e .  Solve the equation:  2 sin 2x · sin 6x = cos 4x.

    S o l u t i o n .  Transform the left-hand side to the sum:

                                        cos 4x – cos 8x = cos 4x ,

                                                  cos 8x = 0 ,

7. Universal substitution. Consider this method by example.

    E x a m p l e .  Solve the equation:  3 sin x – 4 cos x = 3 .

                             So, only the first case gives the solution.