# Trigonometric Equations

Example 1

Find all the angles for which Viewed in a unit circle, sine is the y-value, so at two angles the sine value is 0,5 The first angle at which the sine has a value of 0,5 is 𝜋 / 6, so The period of the sine is 2𝜋, so every full circle the sine has the same value. All solutions to the equation sin (x) = 0,5 are The previous n is an integer. It can be positive or negative, meaning we can rotate the circle in either direction.

Example 2

Solve the equation Trigonometric equations are treated like any other equation. The first step is to get the equation to sin (x) = a From the table we get the first angle from which the equation solves. It is x = 𝜋 / 3 and the other angle is x = 𝜋-𝜋 / 3 = 2𝜋 / 3, so the solutions to the equation are Example 3

Solve the equation The angles at which the sine have these values are 𝜋 / 4 + n2𝜋 and 3𝜋 / 4 + n2𝜋. Then we get When both equations are divided by two, we can solve x where n is an integer.

When solving the equation, it is good to note that the period needs to be divided also. Just like in any equation, all terms are divided.

Example 4

Solve the equation First modify the equation to the form cos (x) = a With a calculator, the angle at which the cosine is 1/3 is 1,2309 .... The solutions of the equation are where n is an integer.

Example 5

Solve the equations in a, the angles must be equal or supplementary. From these equations we can solve the angle x. Move x to the left of the equations. n is an integer and contains both negative and positive integers, so we can give the first answer without a minus sign.

The equation in b can be modified to The tangent is not defined by the angles 𝜋 / 2 + n𝜋, where n is an integer. That is, at the angles where the cosine has a value of 0. The period of the tangent is 𝜋, so the solution of the equation is where n is an integer.

Example 6

Solve the equation Modify the equation so that there is only sine or cosine. According to Pythagorean trigonometric identity We substitute this into our equation And we get a quadratic equation. We denote sin(x)=t This gives sin (x) = 1/2 or sin (x) = 2. The latter is invalid, as the sine only gets values between -1 and 1. Then only sin (x) = 1/2 is valid, so the solutions are where n is an integer.

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