Solve by general equationtana+tan2a+tan3a=0

suraj prem chand
25 Points
8 years ago
tan(2a) = 2 * tan(a) / (1 - tan(a)^2)

tan(3a) =>
tan(2a + a) =>
(tan(a) + tan(2a)) / (1 - tan(a) * tan(2a)) =>
(tan(a) + 2 * tan(a) / (1 - tan(a)^2)) / (1 - tan(a) * 2 * tan(a) / (1 - tan(a)^2)) =>
(tan(a) - tan(a)^3 + 2tan(a)) / (1 - tan(a)^2 - 2tan(a)^2) =>
(3tan(a) - tan(a)^3) / (1 - 3tan(a)^2) =>
tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)

tan(a) + tan(2a) + tan(3a) = 0
tan(a) + 2tan(a) / (1 - tan(a)^2) + tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)) = 0

tan(a) = 0
a = pi * k

1 + 2/(1 - tan(a)^2) + (3 - tan(a)^2) / (1 - 3tan(a)^2) = 0
(1 * (1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2)) / ((1 - tan(a)^2) * (1 - 3tan(a)^2)) = 0

(1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2) = 0
1 - 4tan(a)^2 + 3tan(a)^4 + 2 - 6tan(a)^2 + 3 - 4tan(a)^2 + tan(a)^4 = 0
6 - 14tan(a)^2 + 4tan(a)^4 = 0
2tan(a)^4 - 7tan(a)^2 + 3 = 0
tan(a)^2 = (7 +/- sqrt(49 - 24)) / 4
tan(a)^2 = (7 +/- 5) / 4
tan(a)^2 = 12/4 , 2/4
tan(a)^2 = 3 , 1/2
tan(a) = +/- sqrt(3) , +/- sqrt(2) / 2

tan(a) = 0 , -sqrt(3) , sqrt(3) , -sqrt(2)/2 , sqrt(2)/2tan(2a) = 2 * tan(a) / (1 - tan(a)^2)

tan(3a) =>
tan(2a + a) =>
(tan(a) + tan(2a)) / (1 - tan(a) * tan(2a)) =>
(tan(a) + 2 * tan(a) / (1 - tan(a)^2)) / (1 - tan(a) * 2 * tan(a) / (1 - tan(a)^2)) =>
(tan(a) - tan(a)^3 + 2tan(a)) / (1 - tan(a)^2 - 2tan(a)^2) =>
(3tan(a) - tan(a)^3) / (1 - 3tan(a)^2) =>
tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)

tan(a) + tan(2a) + tan(3a) = 0
tan(a) + 2tan(a) / (1 - tan(a)^2) + tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)) = 0

tan(a) = 0
a = pi * k

1 + 2/(1 - tan(a)^2) + (3 - tan(a)^2) / (1 - 3tan(a)^2) = 0
(1 * (1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2)) / ((1 - tan(a)^2) * (1 - 3tan(a)^2)) = 0

(1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2) = 0
1 - 4tan(a)^2 + 3tan(a)^4 + 2 - 6tan(a)^2 + 3 - 4tan(a)^2 + tan(a)^4 = 0
6 - 14tan(a)^2 + 4tan(a)^4 = 0
2tan(a)^4 - 7tan(a)^2 + 3 = 0
tan(a)^2 = (7 +/- sqrt(49 - 24)) / 4
tan(a)^2 = (7 +/- 5) / 4
tan(a)^2 = 12/4 , 2/4
tan(a)^2 = 3 , 1/2
tan(a) = +/- sqrt(3) , +/- sqrt(2) / 2

tan(a) = 0 , -sqrt(3) , sqrt(3) , -sqrt(2)/2 , sqrt(2)/2tan(2a) = 2 * tan(a) / (1 - tan(a)^2)

tan(3a) =>
tan(2a + a) =>
(tan(a) + tan(2a)) / (1 - tan(a) * tan(2a)) =>
(tan(a) + 2 * tan(a) / (1 - tan(a)^2)) / (1 - tan(a) * 2 * tan(a) / (1 - tan(a)^2)) =>
(tan(a) - tan(a)^3 + 2tan(a)) / (1 - tan(a)^2 - 2tan(a)^2) =>
(3tan(a) - tan(a)^3) / (1 - 3tan(a)^2) =>
tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)

tan(a) + tan(2a) + tan(3a) = 0
tan(a) + 2tan(a) / (1 - tan(a)^2) + tan(a) * (3 - tan(a)^2) / (1 - 3tan(a)^2)) = 0

tan(a) = 0
a = pi * k

1 + 2/(1 - tan(a)^2) + (3 - tan(a)^2) / (1 - 3tan(a)^2) = 0
(1 * (1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2)) / ((1 - tan(a)^2) * (1 - 3tan(a)^2)) = 0

(1 - tan(a)^2) * (1 - 3tan(a)^2) + 2 * (1 - 3tan(a)^2) + (3 - tan(a)^2) * (1 - tan(a)^2) = 0
1 - 4tan(a)^2 + 3tan(a)^4 + 2 - 6tan(a)^2 + 3 - 4tan(a)^2 + tan(a)^4 = 0
6 - 14tan(a)^2 + 4tan(a)^4 = 0
2tan(a)^4 - 7tan(a)^2 + 3 = 0
tan(a)^2 = (7 +/- sqrt(49 - 24)) / 4
tan(a)^2 = (7 +/- 5) / 4
tan(a)^2 = 12/4 , 2/4
tan(a)^2 = 3 , 1/2
tan(a) = +/- sqrt(3) , +/- sqrt(2) / 2

tan(a) = 0 , -sqrt(3) , sqrt(3) , -sqrt(2)/2 , sqrt(2)/2