Which of the following lists contains only functions with vertical asymptotes in their graphs?
A. Cosine, sine, tangent, cotangent
B. Tangent, secant, cosecant, cotangent
C. Sine, tangent, secant, cosecant
D. Cosine, sine, secant, cosecant
cosine, sine, vary between 1, -1? So they cannot be in the correct answer.
To determine which of the given lists contains only functions with vertical asymptotes in their graphs, we need to understand the concept of vertical asymptotes and the characteristics of each function.
A vertical asymptote occurs in the graph of a function when the function approaches positive or negative infinity as the input approaches a specific value. It represents a vertical line that the graph gets arbitrarily close to but never touches.
Now, let's analyze each list of functions:
A. Cosine, sine, tangent, cotangent
The functions cosine and sine are periodic functions and do not have vertical asymptotes. The tangent function has vertical asymptotes wherever its cosine value is zero, which occurs at odd multiples of π/2. The cotangent function, being the reciprocal of the tangent function, also has vertical asymptotes at the same points. Therefore, option A contains functions with vertical asymptotes.
B. Tangent, secant, cosecant, cotangent
The tangent and cotangent functions, as explained above, have vertical asymptotes. However, the secant and cosecant functions are the reciprocals of the cosine and sine functions, respectively, and do not have vertical asymptotes. Thus, option B does not contain only functions with vertical asymptotes.
C. Sine, tangent, secant, cosecant
Similar to option B, the sine function does not have vertical asymptotes. The tangent function has vertical asymptotes, but the secant and cosecant functions do not. Therefore, option C does not contain only functions with vertical asymptotes.
D. Cosine, sine, secant, cosecant
Both cosine and sine functions are periodic and do not have vertical asymptotes. The secant and cosecant functions, being the reciprocals of cosine and sine, respectively, do have vertical asymptotes. Therefore, option D contains only functions with vertical asymptotes.
In conclusion, the correct list of functions containing only functions with vertical asymptotes is option D: Cosine, sine, secant, and cosecant.