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Solve by general equation tana+tan2a+tan3a=0

Solve by general equation
tana+tan2a+tan3a=0

Grade:10

1 Answers

suraj prem chand
25 Points
5 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

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