Can you explain why this is the answer
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Can you explain why this is the answer

[From: ] [author: ] [Date: 12-01-07] [Hit: ]
and pi/2)tan^2 3x = 3tan 3x = +/- sqrt(3)tan u = +/- sqrt(3) at u = pi/3 , 2pi/3 , 4pi/3, and 5pi/3 (where y = +/- sqrt(3)/2 and x = +/- 1/2)so the general solution is u = pi/3 + n * pi or u = 2pi/3 + n * pi (where n is an integer)here, u = 3xso one series of solutions is 3x = pi/3 + n * pi ==> x = pi/9 + n * pi/3and the other is 3x = 2pi/3 + n * pi ==> x = 2pi/9 + n * pi/3these are the general solutionsfor the 12 specific solutions between 0 and 2pi, let n = 0 to 5n = 0 ==> x = pi/9 or 2pi/9n = 1 ==> x = 4pi/9 or 5pi/9 (note that pi/3 = 3pi/9)n = 2 ==> x = 7pi/9 or 8pi/9n = 3 ==> x =10pi/9 or 11pi/9n = 4 ==> x = 13pi/9 or 14pi/9n = 5 ==> x = 16pi/9 or 17pi/9n = 6 will take us out of the interval 0 to 2piso 12 total solutions:n = pi/9,......
multiples of 30 , 45, and 60 (pi/6 , pi/4, and pi/2)

tan^2 3x = 3
tan 3x = +/- sqrt(3)

tan u = +/- sqrt(3) at u = pi/3 , 2pi/3 , 4pi/3, and 5pi/3 (where y = +/- sqrt(3)/2 and x = +/- 1/2)

so the general solution is u = pi/3 + n * pi or u = 2pi/3 + n * pi (where n is an integer)

here, u = 3x
so one series of solutions is 3x = pi/3 + n * pi ==> x = pi/9 + n * pi/3
and the other is 3x = 2pi/3 + n * pi ==> x = 2pi/9 + n * pi/3

these are the general solutions

for the 12 specific solutions between 0 and 2pi, let n = 0 to 5

n = 0 ==> x = pi/9 or 2pi/9
n = 1 ==> x = 4pi/9 or 5pi/9 (note that pi/3 = 3pi/9)
n = 2 ==> x = 7pi/9 or 8pi/9
n = 3 ==> x = 10pi/9 or 11pi/9
n = 4 ==> x = 13pi/9 or 14pi/9
n = 5 ==> x = 16pi/9 or 17pi/9

n = 6 will take us out of the interval 0 to 2pi

so 12 total solutions: n = pi/9, 2pi/9 , 4pi/9 , 5pi/9 , 7pi/9 , 8pi/9 , 10pi/9 , 11pi/9 , 13pi/9 , 14pi/9 , 16pi/9, or 17pi/9

there are 2 solutions per period, and since the period of tan x is pi ==> the period of tan 3x is pi/3, there are 6 periods of tan 3x between 0 and 2pi

-
tan²(3x) = 3

tan (3x) = ±√3

Solutions:
x = 1/9(3.π.n - π),
x = 1/9(3.π.n + π),

where n is a member of the set of integers

-
Tan x = -1

Since answer is - (Negative) angle is in Quadreants 2 and 4
where the tangent function is negative.

What angle has a tangent of +1....45 degrees or pi/4

Now put that angle in Quadrants 2 (3pi/4) qnd 4 (7pi/4)


tan² x = 3

√(tan² x) = +- √3

angle with tangent √3 is 60 degrees or pi/3

pi/3....2pi/3.....4pi/3.....5pi/3 for answer 0<= x < 2pi

For All answer......pi/3 + n*pi......2pi/3 + n*pi

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