**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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