Solve (0<=©<=360). ©Represents theta
tan©+tan2©+tan3©=0
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https://answers.yahoo.com/question/index?qid=20061106084200AAifTZM
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To solve the equation tan© + tan2© + tan3© = 0 where © represents theta, we can start by using the identity tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B)).
Let's simplify the equation by applying the identity twice:
tan© + tan2© + tan3© = 0
Using the identity, we can rewrite tan2© as tan(© + ©) and tan3© as tan(© + © + ©):
tan© + (tan© + tan©)/(1 - tan©tan©) + (tan(© + ©) + tan©)/(1 - tan(© + ©)tan©) = 0
Now we can simplify further:
tan© + (2tan©)/(1 - tan2©) + (tan(© + ©) + tan©)/(1 - tan(© + ©)tan©) = 0
Next, use the angle addition formula for tan(A + B):
tan(A + B) = (tan(A) + tan(B))/(1 - tan(A)tan(B))
Applying the formula to the equation:
tan© + (2tan©)/(1 - tan2©) + (tan© + tan©)/(1 - tan(© + ©)tan©) = 0
The equation now becomes:
tan© + (2tan©)/(1 - tan2©) + (tan© + tan©)/(1 - (tan© + ©)/(1 - tan©tan©)tan©) = 0
Now we can simplify and combine the terms:
tan© + (2tan©)/(1 - tan2©) + (2tan©)/(1 - (tan© + ©)/(1 - tan©tan©)tan©) = 0
Multiplying through by the common denominators to eliminate fractions:
(tan©(1 - tan2©) + 2tan2© + 2tan©(1 - tan©tan©tan©))/(1 - tan2©)(1 - (tan© + ©)/(1 - tan©tan©)tan©) = 0
Now we can simplify further:
(tan© - tan3© + 2tan2© + 2tan© - 2tan3© + 2tan©)/(1 - tan©tan© - (tan© + ©)tan© + (tan© + ©))/(1 - tan©tan©tan© - (tan© + ©)tan©) = 0
Combining like terms:
(3tan© + 2tan2© - 3tan3©)/(1 - 2tan© - ©tan©) = 0
At this point, we have simplified the equation using trigonometric identities, but finding the exact solutions for theta (©) requires additional steps.
To solve the equation further and find the solutions for ©, we can apply numerical methods like Newton's method, which involves iterative approximation, or we can use graphing software to visualize the equation and find the values where the graph intersects the x-axis.
Note: Since the equation involves trigonometric functions, there might be infinitely many solutions for © within the given range (0 <= © <= 360).