explore graphing the basic tangent functions\nread explore and complete the following activity (adapted from…

explore graphing the basic tangent functions\nread explore and complete the following activity (adapted from lesson 18.1).\nrecall that the tangent of an angle can be found from the relationship tan θ = sin θ / cos θ. using the coordinates of the position of the unit - circle, (x,y)=(cos θ,sin θ), the value of the tangent function can also be found from the ratio of y to x.\nin other words, recall tan θ = y / x\ntry this - use your knowledge of the tangent function and the unit circle to complete the following.\nradian tan θ\n-π/2 undefined\n-π/4 - 1\n0 0\nπ/4 1\nπ/2 undefined\nobserve the reference points from above to its graph function f(θ)=tan θ\nwhy are there asymptotes in the function f(θ)=tan θ?\nthe tangent function has asymptotes where the function is undefined.

explore graphing the basic tangent functions\nread explore and complete the following activity (adapted from lesson 18.1).\nrecall that the tangent of an angle can be found from the relationship tan θ = sin θ / cos θ. using the coordinates of the position of the unit - circle, (x,y)=(cos θ,sin θ), the value of the tangent function can also be found from the ratio of y to x.\nin other words, recall tan θ = y / x\ntry this - use your knowledge of the tangent function and the unit circle to complete the following.\nradian tan θ\n-π/2 undefined\n-π/4 - 1\n0 0\nπ/4 1\nπ/2 undefined\nobserve the reference points from above to its graph function f(θ)=tan θ\nwhy are there asymptotes in the function f(θ)=tan θ?\nthe tangent function has asymptotes where the function is undefined.

Answer

Explanation:

Step1: Recall tangent - cosine relationship

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, the function is undefined when $\cos\theta = 0$.

Step2: Identify $\theta$ values for $\cos\theta=0$

The cosine function $\cos\theta = 0$ when $\theta=\frac{\pi}{2}+k\pi$, where $k\in\mathbb{Z}$ (set of all integers). At these values of $\theta$, the tangent function $\tan\theta=\frac{\sin\theta}{\cos\theta}$ has a division - by - zero, making it undefined.

Step3: Define asymptotes

In the graph of a function, vertical asymptotes occur at the values of the independent variable for which the function is undefined. So, the vertical asymptotes of $y = \tan\theta$ are at $\theta=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$.

Answer:

The tangent function $f(\theta)=\tan\theta$ has asymptotes at $\theta=\frac{\pi}{2}+k\pi$, where $k$ is an integer because $\tan\theta=\frac{\sin\theta}{\cos\theta}$ and $\cos\theta = 0$ at these values, causing the function to be undefined.