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:
cos2 x – sin2 x = sinx

2. Use the formulas to get only one trigonometric ratio:
1 – sin2 x - sin2 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) 2sin4x - 7cos2x + 3 = 0




a) x €{45o + k90o, k€Ζ}

b) x €{k90o, k€Ζ}, x €{120o + k360o, k€Ζ}, x €{240o + k360o, k€Ζ}

c) x €{45o + k360o, k€Ζ}, x €{135o + k360o, k€Ζ}

d) x €{45o + k90o, k€Ζ}

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