Trigonometric Equations

In this lesson, we will look at solving Trigonometric Equations.

A trigonometric equation is one that states a relation between trigonometric functions of unknown angles (or numbers)

Trigonometric equations, in general, have an unlimited number of solutions. Usually the domain is restricted to 0 le theta le 360, to limit the number of solutions.

No general method for solving equations can be given. However, the following suggestions will be helpful.

1) Reduce the equation to a simpler equation by factoring, if possible.
2) Simplify the functions of different angles to functions of the same angle by means of known formulas.
3) Simplify the equation so that it involves only the same function of the angle.
4) Check your answers.

Example:

Solve
2 cos² x = – 3 cos x + 2 for x, 0° ≤ x < 360°

Solution:

2 cos² x = – 3 cos x + 2
2 cos² x + 3 cos x – 2 = 0
(2 cos x – 1)(cos x + 2) = 0

2 cos x – 1 = 0

x = 60°, 300°

or

cos x + 2 = 0
cos x = –2
x = Ø

Therefore, x = 60°, 300°

Example:

Solve sin 2x = 3 sin x for x, 0° ≤ x < 360°

Solution:

sin 2x = 3 sin x
2 sin x cos x = 3 sin x
2 sin x cos x – 3 sin x = 0
sin x (2 cos x – 3) = 0

sin x = 0
x = 0°, 180°

or

2 cos x – 3 = 0

Therefore, x = 0° or 180°