Math

Question Identify the trig function with vertical asymptotes at 0,π,2π0, \pi, 2\pi and period 2π2\pi.

Studdy Solution

STEP 1

Assumptions
1. Vertical asymptotes occur where the function is undefined and the function values approach infinity.
2. A period of a function is the smallest interval over which the function repeats itself.
3. The trigonometric functions in question are cotx\cot x, tanx\tan x, secx\sec x, and cscx\csc x.
4. The values 00, π\pi, and 2π2\pi are specific points where we are checking for vertical asymptotes.

STEP 2

Recall the properties of the tangent function, y=tanxy = \tan x, which is undefined at x=π2+kπx = \frac{\pi}{2} + k\pi for any integer kk. This is where its vertical asymptotes are located.

STEP 3

Determine the period of the tangent function. The tangent function has a period of π\pi, meaning it repeats every π\pi units.

STEP 4

Since the period of tanx\tan x is π\pi, it will not have vertical asymptotes at 00, π\pi, and 2π2\pi simultaneously within one period. Therefore, y=tanxy = \tan x does not meet the criteria.

STEP 5

Recall the properties of the cotangent function, y=cotxy = \cot x, which is undefined at x=kπx = k\pi for any integer kk. This is where its vertical asymptotes are located.

STEP 6

Determine the period of the cotangent function. The cotangent function has a period of π\pi, meaning it repeats every π\pi units.

STEP 7

Since the period of cotx\cot x is π\pi, it will have vertical asymptotes at 00, π\pi, and 2π2\pi within one period of 2π2\pi. Therefore, y=cotxy = \cot x meets the criteria.

STEP 8

Recall the properties of the secant function, y=secxy = \sec x, which is undefined at x=π2+kπx = \frac{\pi}{2} + k\pi for any integer kk. This is where its vertical asymptotes are located.

STEP 9

Determine the period of the secant function. The secant function has a period of 2π2\pi, meaning it repeats every 2π2\pi units.

STEP 10

Since the period of secx\sec x is 2π2\pi, it will not have vertical asymptotes at 00, π\pi, and 2π2\pi simultaneously within one period. Therefore, y=secxy = \sec x does not meet the criteria.

STEP 11

Recall the properties of the cosecant function, y=cscxy = \csc x, which is undefined at x=kπx = k\pi for any integer kk. This is where its vertical asymptotes are located.

STEP 12

Determine the period of the cosecant function. The cosecant function has a period of 2π2\pi, meaning it repeats every 2π2\pi units.

STEP 13

Since the period of cscx\csc x is 2π2\pi, it will have vertical asymptotes at 00, π\pi, and 2π2\pi within one period of 2π2\pi. Therefore, y=cscxy = \csc x meets the criteria.

STEP 14

Based on the analysis, the functions that have vertical asymptotes at 00, π\pi, and 2π2\pi, and a period of 2π2\pi are:
a) y=cotxy = \cot x d) y=cscxy = \csc x
These functions meet both criteria of having the specified vertical asymptotes and the period of 2π2\pi.

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