For what numbers x 2 pi lessthanorequalto x lessthanorequalt

For what numbers x, -2 pi lessthanorequalto x lessthanorequalto 2pi, does the graph of y = tan x have vertical asymptotes? Refer to the graph, which shows y = tan x in blue and y = cos x in red. In the interval -2pi lessthanorequalto x lessthanorequalto 2pi, what is the smallest value of x for which the graph of y = tan x has a vertical asymptote? (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression. Simplify your answer.) Within the given interval, what is the next x-value for which y = tan x has a vertical asymptote? (Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression. Simplify your answer.) At what other x-values in the interval - 2 pi lessthanorequalto x lessthanorequalto 2pi does the graph of y = tan x have a vertical asymptote? pi/2, and 3pi/2 There are no others. 3pi/2 and 5 pi/2 pi/2, 3pi/2, and 5pi/2

Solution

the x intercepts of the graph y = cos x are the vertical asymptote of y = tan x

x intercepts of cos x are -3pi/2 , -pi/2 , + pi/2 , 3pi/2

the smallest value of x for which y = tan x has vertical asymptote is

x = -3pi/2

the next value for which y = tan x has vertical asymptote is

x = -pi/2

other values in the interval -2pi <= x <= 2pi

the x values for which function has vertical asymptotes are

x = pi/2 and 3pi/2