# Graph of f(x) = cosx

We will first begin to graph the function, f(x) = cosx by creating a table of values. For our extensive purposes, we will graph this function over the interval of [0, 2π], or simply, 1 rotation. We will use our values from the unit circle derivation for these points:

x | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π | 7π/6 | 5π/4 | 4π/3 | 3π/2 | 5π/3 | 7π/4 | 11π/6 | 2π |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

f(x) | 1 | √3/2 | √2/2 | 1/2 | 0 | -1/2 | -√2/2 | -√3/2 | -1 | -√3/2 | -√2/2 | -1/2 | 0 | 1/2 | √2/2 | √3/2 | 1 |

When we graph this obtain:

## Domain and Range of f(x) = sinx

The function f(x) = cosx extends from -∞ to ∞ in the x-direction. For example, we can let x = -100000 and evaluate f(-100000) to obtain a value. The y-direction of the function f(x) is restricted between -1 to 1 inclusive though.

(1)And:

(2)Or in domain and range interval notation:

(3)# Graph of f(x) = sinx

We will graph f(x) = sinx in the same manner as we did f(x) = cosx. The table of values below are with regards to f(x) = sinx.

x | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π | 7π/6 | 5π/4 | 4π/3 | 3π/2 | 5π/3 | 7π/4 | 11π/6 | 2π |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

f(x) | 0 | 1/2 | √2/2 | √3/2 | 1 | √3/2 | √2/2 | 1/2 | 0 | -1/2 | -√2/2 | -√3/2 | -1 | -√3/2 | -√2/2 | -1/2 | 0 |

The graph below is of f(x) = sinx:

## Domain and Range of f(x) = sinx

The function f(x) = sinx extends from -∞ to ∞ in the x-direction as well. For example, we can let x = 345345 and evaluate f(345345) to obtain a value. The y-direction of the function f(x) is restricted between -1 to 1 inclusive though as well.

(4)And:

(5)Or in domain and range interval notation:

(6)# Graph of f(x) = tanx

We can clearly see that the functions f(x) = cosx and f(x) = sinx look rather similar. Unfortunately, this is not the case for f(x) = tanx. Let's first look at the table of values associated with f(x) = tanx:

x | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π | 7π/6 | 5π/4 | 4π/3 | 3π/2 | 5π/3 | 7π/4 | 11π/6 | 2π |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

f(x) | 0 | 1/√3 | 1 | √3 | d.n.e. | -√3 | -1 | -1/√3 | 0 | 1/√3 | 1 | √3 | d.n.e. | -√3 | -1 | -1/√3 | 0 |

Unfortunately, we have values of the function that **do not exist** (*abbreviated d.n.e.*). At these points, we have vertical asymptotes for the graph. In the case of f(x) = tanx, the equations of our vertical asymptotes are as follows:

We know however that the domain of the function f(x) = tanx exists from -∞ to ∞, while the range of the functino f(x) exists from -1 to 1. Thus we can say that:

The graph below is of f(x) = tanx:

## Domain and Range of f(x) = tanx

In the case of f(x) = tanx in the x-direction, we have some inconsistencies. From the table of values above, we know that x cannot equal π/2 , or 3π/2 as f(π/2) and f(3π/2) will result in an undefined result. In fact, x = π/2 , 3π/2 , 5π/2 , 7π/2 , etc… cannot be evaluated for f(x). We can write a general equation for these inconsistencies in the domain of tanx by writing a general rule for the asymptotes of our function

(8)For any value of k that exists as an integer {… -2, -1, 0, 1, 2, …} we will obtain an equation of a vertical asymptote for the graph f(x) = tanx. Thus we can say our domain of tanx to be:

(9)Additionally, as we can see from the graph of f(x) = tanx, the range of the function will extend from negative infinity to positive infinity. We can thus denote the range of tanx as:

(10)