Given that tan(θ)=5, what is tan(2θ)?

Answer 1

Hi Mikaela,

I think of tangent as a graph that exists between two boundaries pi/2 to -pi/2 and it repeats every pi times. So for the first part θ =78º does exist between the boundaries of 90º and -90º. So 2*78º = 156º. I like to think of this as a slight shift in the function in the original domain between pi/2 and -pi/2. you can shift the function back to this location by subtracting 180º to get -24º. I am sure there is a more maths way to describe this using "n's" or "k's" but this is how I have understood this function.

to clarify I am stating that tan(-24º) = tan(156º) = -0.445 . We are just cycling through the same plot reaching the end and going back to the beginning.

I hope this helps!

Answer 2

The double angle identity for tangent is:

So tan(2θ) = 2(5)/(1 - 5²) = 10/-24 = -5/12

Answer 3

tan2x = 2tanx/(1-tan^2x)

= 2(5)/(1-25) = 10/-24 = - 5/12

another way to do this is

use a calculator and find x for tanx=5. find the inverse tangent of 5 = 78.7 degrees

double it to get 157.4 degrees, then find the tangent of 157.4 = -0.42 which is approximately - 5/12