Question

where does the graph of the function $y = \tan(x)$ have asymptotes?

• at the values of $x$ where $\cos(x) = 0$
• at the values of $x$ where $\sin(x) = 0$
• at the values of $x$ where $\cos(x)$ is undefined
• at the values of $x$ where $\sin(x)$ is undefined

Explanation:

Step1: Recall the definition of tangent function

The tangent function is defined as \(\tan(x)=\frac{\sin(x)}{\cos(x)}\). A vertical asymptote occurs where the function is undefined, which for a rational function like \(\tan(x)\) is where the denominator is zero (and the numerator is not zero at those points).

Step2: Analyze when the denominator is zero

For \(\tan(x)=\frac{\sin(x)}{\cos(x)}\), the denominator is \(\cos(x)\). So, \(\tan(x)\) is undefined when \(\cos(x) = 0\) (since \(\sin(x)\) is defined for all real \(x\) and at the points where \(\cos(x)=0\), \(\sin(x)=\pm1
eq0\)).

Now let's analyze each option:

• Option 1: "at the values of \(x\) where \(\cos(x) = 0\)" – This matches our analysis because \(\tan(x)\) is undefined (has vertical asymptotes) when \(\cos(x)=0\).
• Option 2: "at the values of \(x\) where \(\sin(x)=0\)" – When \(\sin(x)=0\), \(\tan(x)=\frac{0}{\cos(x)} = 0\) (as long as \(\cos(x)

eq0\)), so there is no asymptote here.

• Option 3: "at the values of \(x\) where \(\cos(x)\) is undefined" – But \(\cos(x)\) is defined for all real \(x\), so this is incorrect.
• Option 4: "at the values of \(x\) where \(\sin(x)\) is undefined" – \(\sin(x)\) is defined for all real \(x\), so this is incorrect.

Answer:

A. at the values of \(x\) where \(\cos(x) = 0\)