In this question, we need to identify which trigonometric functions have asymptotes.

Asymptotes occur in trigonometric functions when the function approaches infinity or negative infinity for certain values of $x$. Here's an analysis of each function:

1. Sine and Cosine: These functions are continuous and oscillate between -1 and 1 without approaching infinity, so they do not have asymptotes.

2. Tangent: The tangent function has asymptotes where $\cos(x) = 0$, such as at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots$. Therefore, tangent has asymptotes.

3. Cosecant: The cosecant function is the reciprocal of sine ($\csc(x) = \frac{1}{\sin(x)}$). It has asymptotes where $\sin(x) = 0$, such as at $x = 0, \pi, 2\pi, \dots$. Thus, cosecant has asymptotes.

4. Secant: The secant function is the reciprocal of cosine ($\sec(x) = \frac{1}{\cos(x)}$). It has asymptotes where $\cos(x) = 0$, such as at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots$. Hence, secant has asymptotes.

5. Cotangent: The cotangent function has asymptotes where $\sin(x) = 0$, such as at $x = 0, \pi, 2\pi, \dots$. Therefore, cotangent has asymptotes.

Answer

The trigonometric functions that have asymptotes are:

• Tangent
• Cosecant
• Secant
• Cotangent

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Related Questions

1. Why do the tangent, cosecant, secant, and cotangent functions have asymptotes while sine and cosine do not?
2. How can we determine the locations of asymptotes in trigonometric functions graphically?
3. What are the equations of the vertical asymptotes for the tangent function?
4. How does the period of a trigonometric function affect the placement of its asymptotes?
5. Are there other types of asymptotes (e.g., horizontal or slant) in trigonometric functions?

Tip

To find asymptotes for reciprocal trigonometric functions, look for values where the original sine or cosine functions are zero, as this causes the reciprocal functions to diverge to infinity.