Find tan(3 theta) in terms of tan theta
To find tan(3theta) in terms of tan(theta), we can use the triple angle formula for tangent. The formula is as follows:
tan(3theta) = (3tan(theta) - tan^3(theta)) / (1 - 3tan^2(theta))
Now let's go through how to derive the triple angle formula for tangent:
Step 1: Start with the double angle formula for tangent:
tan(2theta) = (2tan(theta)) / (1 - tan^2(theta))
Step 2: Substitute theta with 2theta in the double angle formula:
tan(2(2theta)) = (2tan(2theta)) / (1 - tan^2(2theta))
tan(4theta) = (2tan(2theta)) / (1 - tan^2(2theta))
Step 3: Rewrite tan(2theta) in terms of tan(theta):
tan(2theta) = (2tan(theta)) / (1 - tan^2(theta))
Step 4: Substitute tan(2theta) in the equation from Step 2:
tan(4theta) = (2(2tan(theta)) / (1 - 4tan^2(theta) / (1 - tan^2(theta))
tan(4theta) = (4tan(theta) - 4tan^3(theta)) / (1 - 6tan^2(theta) + tan^4(theta))
Step 5: Simplify the expression by factoring (1 - tan^2(theta)):
tan(4theta) = (4tan(theta) - 4tan^3(theta)) / ((1 - tan^2(theta))(1 + tan^2(theta)))
tan(4theta) = (4tan(theta) - 4tan^3(theta)) / (1 - tan^2(theta))(sec^2(theta))
Step 6: Notice that (1 - tan^2(theta))(sec^2(theta)) is equal to 1.
tan(4theta) = (4tan(theta) - 4tan^3(theta))
Now we have derived the formula for tan(4theta) in terms of tan(theta). To find tan(3theta) in terms of tan(theta), we can use the fact that 3theta is equal to 4theta - theta:
tan(3theta) = tan(4theta - theta) = (tan(4theta) - tan(theta)) / (1 + tan(4theta)tan(theta))
= ((4tan(theta) - 4tan^3(theta)) - tan(theta)) / (1 + (4tan(theta) - 4tan^3(theta))tan(theta))
= (4tan(theta) - 4tan^3(theta) - tan(theta)) / (1 + 4tan^2(theta) - 4tan^4(theta) + 4tan^2(theta) - 4tan^4(theta))
= (3tan(theta) - 4tan^3(theta)) / (1 - 6tan^2(theta) + 4tan^4(theta))
Therefore, tan(3theta) = (3tan(theta) - 4tan^3(theta)) / (1 - 6tan^2(theta) + 4tan^4(theta)), which is the triple angle formula for tangent.