Find all solutions of the equation tan^5𝑥−9tan𝑥=0. The answer is 𝐴𝑘𝜋 where 𝑘 is any integer,

Question

I thought it was sqrt3 but that was incorrect, I also tried pi/3

Answer 1

tan^5(x) - 9tan(x) = 0

factor out tan(x)

tan(x)[tan^4(x) - 9] = 0

set each factor = 0

tan(x) = 0, x= kpi where k= any integer

or x = 180k where k= any integer

= 0, 180, 360, 540 ... degrees

or -180, -360, -540 etc.

tan^4(x) -9 =0

tan^4(x) = 9

tan^2(x) = 3

tan(x) = sqr3 or -sqr3

x = 60 degrees + 180k where k = any integer

or 120 degrees + 180k where k = any integer

or pi/3 + kpi where k = any integer

or 2pi/3 +kpi where k= any integer

some people us n instead of k. n tends to mean natural numbers or whole numbers.

Answer 2

tan^5𝑥−9tan𝑥=tan(x)(tan^4-9)=tan(x)(tan^2(x)-3)(tan^2(x)+3)=0

tan(𝑥)=0

tan^-1(tan(x))=tan^-1(0)

x=0,pi

(tan^2(x)-3)=0

tan^2(x)=3

tan(x)=sqrt(3)

tan^-1(tan(x))=tan^-1(sqrt(3))

x= (kpi)/3