Find the domain of the function y = 5∕3 tan(3∕4x). A)  All real numbers except odd integer multiples of 2π∕3
B)  All real numbers except odd integer multiples of 4π∕3
C)  All real numbers except 0 and odd integer multiples of 2π∕3
D)  All real numbers except 0 and odd integer multiples of 4π∕3​

A

Step-by-step explanation:

tan(3x/4)=sin(3x/4)/cos(3x/4)

So the domain of tah(3x/4) is all real numbers except real numbers that make cos(3x/4)=0.

cos(pi/2 +n pi)=0

So we need to solve 3x/4=pi/2+n pi

Multiply both sides by 4/3: x=4/3(pi/2+n pi)

Distribute: x=2pi/3+4n pi/3

Or x=(2pi+4 n pi)/3

Or x=2 pi/3 ×(1+2n)

So odd integer multiples of 2pi/3 is the numbers to be excluded from the domain.