**Trigonometric functions**

Already, with the help of the unit circle it was noticed that sine and cosine get values between [-1,1]. If sine and cosine are defined as functions sin(x) and cos(x), then the set of values of these functions is [-1,1]

**Graph of f(x)=sin(x)**

**Graph of f(x)=cos(x)**

The period of both sine and cosine is *2π*, the values of the function are repeated every* 2π.*

The figure is a graph of* f(x) = sin(x)*. The main period is drawn in red, that is the interval* [0,2π]*. After this, the values are repeated between* [2π, 4π]* and continue to repeat every *2π*. It can be seen from the graph that the sine gets the value 0 at *π *and* 2*π. The value of sine is 1 at *π / 2* and* 3π / 2*

The values of the graph correspond to the values of the circumference points in the unit circle.

Sini and cosine get all values between [-1,1]. If we multiply a function by a number, it changes the set of values of the function. Below are graphs of functions that have been multiplied with some numbers

**f (x) = 2sin (x)**

If sine is multiplied by 2, then the value set of the function is [-2,2]

**f (x) = 3sin (x)**

If sine is multiplied by 3, then the value set of the function is [-3,3]

**f (x) = 0,5sin (x)**

If sine is multiplied by 0,5, then the value set of the function is [-0,5;0,5]

If the coefficient is in a variable, it affects the period of the function.

**f (x) = sin (2x)**

By multiplying the variable by 2, the period is halved.

Now the values are repeated every π.

**f (x) = sin (0,5x)**

By multiplying the variable by 0,5, the period is doubled.

Now the values are repeated every 4π.

The period of the tangent is π. The tangent was defined as sine divided by cosine, so the tangent is not defined at cosine zeros. The graph of the tangent is discontinuous, and as we approach the zero point of the cosine, the values of the tangent grow to infinity. The tangent has neither a maximum nor a minimum value. The set of values of the tangent is the whole set of real numbers.

**f (x) = tan (x)**

The values of the tangent are repeated every π. This is the basic period of the tangent.

**f (x) = tan (x) and g (x) = cos (x)**

The tangent is not defined at cosine zeros

**Example 1**

Find the value set and base period of the function

It is known that* sin(x) *gets values in the range *[-1,1]* and *sin(2x)* also gets values in the same range. So

The value set of the function is [-2,2]

The basic period of sine is 2π. The sine function gets the same values every 2π. So

From this we can see that

The basic period of the function *f(x)* is π.

**Example 2**

Find the value set and base period of the function

The set of values of the function *cos(x)* is* [-1,1],* as is the value of the function *cos(3x)*. Let's find out the set of values

The value set of the function* f(x)* is *[-1,5]*

The base period of cosine is 2π, so

From this we can see that

The basic period of the function* f(x) *is *2π / 3*

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