# Trigonometric equations

A **trigonometric equation** is an equation with an unknown in a trigonometric ratio.

For example: cos2x = sinx

To solve it, we have to follow these steps:

1. Apply the formulas and transformations to leave only one angle:

cos^{2} x – sin^{2} x = sinx

2. Use the formulas to get only one trigonometric ratio:

1 – sin^{2} x - sin^{2} x = sinx

3. Solve the equation as if the trigonometric ratio was the unknown:

4. Calculate the angle with the help of a picture:

5. Write the solutions by adding a whole number of circumferences. If the angle is a function of x, work out the unknown.

**Exercise**: Solve the following equations:

a) sin 2x = tanx

b) sinx + sin2x + sin3x = 0

c) tanx · secx = √2

d) 2sin^{4}x - 7cos^{2}x + 3 = 0

Solutions:

a) x €{45^{o} + k90^{o}, k€Ζ}

b) x €{k90^{o}, k€Ζ}, x €{120^{o} + k360^{o}, k€Ζ}, x €{240^{o} + k360^{o}, k€Ζ}

c) x €{45^{o} + k360^{o}, k€Ζ}, x €{135^{o} + k360^{o}, k€Ζ}

d) x €{45^{o} + k90^{o}, k€Ζ}

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