Solving Trigonometric Equations

Solving Trigonometric Equations

How to solve trigonometric equations problems: examples and their solutions.

Example 1

Solve the given equation. cos^2 x - sin x + 1 = 0 (0 <= x <= 2pi)

To factor the left side of the equation,
change the given equation
by using the identities you've learned.

Change cos2 x into 1 - sin2 x.

Pythagorean identities

Think 'sin x' as a variable
and solve the equation by factoring.

Then 'sin x - 1 = 0' or 'sin x + 2 = 0'.

Solving a quadratic equation by factoring

Case 1) sin x - 1 = 0

sin x = 1

So draw y = sin x (0 ≦ x ≦ 2π)
and y = 1
to find the intersecting point.

x = π/2 is the intersecting point.

So x = π/2 is the answer of case 1.

Graphing sine functions

Case 2) sin x + 2 = 0

sin x = -2

But -1 ≦ sin x ≦ 1.

So there's no solution in case 2.

So x = π/2 is the answer.

Example 2

Solve the given equation. sin 2x = sin x (0 <= x <= 2pi)

Move sin x to the left side.

Change sin 2x into 2 sin x cos x.

sin 2A (double-angle formula)

sin x is the GCF.

So sin x(2 cos x - 1) = 0.

So 'sin x = 0' or '2 cos x - 1 = 0'

Solving a quadratic equation by factoring

Case 1) sin x = 0

So draw y = sin x (0 ≦ x ≦ 2π)
and y = 0
to find the intersecting points.

x = 0, π, 2π are the intersecting points.

So x = 0, π, 2π are the answers of case 1.

Graphing sine functions

Case 2) 2 cos x - 1 = 0

cos x = 1/2

So draw y = cos x (0 ≦ x ≦ 2π)
and y = 1/2
to find the intersecting points.

(Think of a right triangle whose cosine is 1/2:
30-60-90 triangle → cos π/3 = 1/2)

So x = π/3, 2π - π/3 are the intersecting points.

So x = π/3, 5π/3 are the answers of case 2.

Graphing cosine functions

So x = 0, π/3, π, 5π/3, 2π are the answers.

Example 3

Solve the given equation. cos 2x - 5 cos x + 3 = 0 (0 <= x <= 2pi)

Change cos 2x into 2 cos2 x - 1.

cos 2A (double-angle formula)

Think 'cos x' as a variable
and solve the equation by factoring.

Then '2 cos x - 1 = 0' or 'cos x - 2 = 0'.

Solving a quadratic equation by factoring

Case 1) 2 cos x - 1 = 0

cos x = 1/2

So draw y = cos x (0 ≦ x ≦ 2π)
and y = 1/2
to find the intersecting points.

(Think of a right triangle whose cosine is 1/2:
30-60-90 triangle → cos π/3 = 1/2)

So x = π/3, 2π - π/3 are the intersecting points.

So x = π/3, 5π/3 are the answers of case 1.

Graphing cosine functions

Case 2) cos x - 2 = 0

cos x = 2

But -1 ≦ cos x ≦ 1.

So there's no solution in case 2.

So x = π/3 and 5π/3 are the answers.