Graphs Of Cosine Sine And Tangent

# 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 f(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 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.

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And:

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\begin{align} -1 \leq f(x) \leq 1 \end{align}

Or in domain and range interval notation:

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# 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 f(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 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.

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\begin{align} -\infty < x < \infty \quad or \quad x \in \mathbb{R} \end{align}

And:

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\begin{align} -1 \leq f(x) \leq 1 \end{align}

Or in domain and range interval notation:

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# 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 f(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 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:

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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

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\begin{align} \quad x \neq \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, ... (2k - 1)\frac{\pi}{2}, \quad k \in \mathbb{I} \end{align}

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:

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