Introducing an auxiliary angle method to solve the trigonometric equation

Trigonometry

Trigonometric Equations

Equations of the Type a · cos x + b · sin x = c

Trigonometric equations examples

Equations of the type a cos x + b sin x = c

To solve the trigonometric equations which are linear in sin x and cos x, and where, a, b, and c are real

numbers we can use the two methods,

a) introducing an auxiliary angle, and b) introducing new unknown.

a) Introducing an auxiliary angle method

Consider the constants a and b as rectangular coordinates of a point expressed by polar coordinates (r, j),

then, a = r cos j and b = r sin j,

By substituting for a and b in the given equation

a · cos x + b · sin x = c

obtained is, r cos x · cos j + r sin x · sin j = c or

using addition formula yields,

since

obtained is

the basic trigonometric equation whose solution is known.

Note that the given equation, a cos x + b sin x = c will have a solution if

it follows that the constants, a, b and c should satisfy relation c² < a² + b².

Introducing an auxiliary angle method example

Example: Solve the equation, sin x + Ö3 · cos x = 1.

Solution: Comparing corresponding parameters of the given equation with a cos x + b sin x = c it follows,

a = Ö3, b = 1 and c = 1.

By substituting given quantities to the basic equation

or x - 30° = + 60° + k · 360° thus, solutions are, x = 90° + k · 360° and x′ = - 30° + k · 360°, kÎ Z.

The same solution can be obtained using following procedure, from sin x + Ö3 · cos x = 1

and a cos x + b sin x = c | ¸ b

that means that we can introduce an auxiliary angle j that is

sin x · sin 30° + cos x · cos 30° = sin 30°, cos (x - 30°) = 1/2

and this is the same basic equation obtained above.

Pre-calculus contents G