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 2π. 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
Turn on the subtitles if needed