Question

Calculate: cos(2arctan(1/2))
[Without the use of a calculator. :-) ]

Answer 1

like a first step write

arctan = arcsin / arccos

Answer 2

Does that relation hold?

Answer 3

hope you know that

tan x = sin x / cos x

and cot x = cos x / sin x

yes ?

Answer 4

Indeed

Answer 5

arctan(1/2) = a; tan(a) = 1/2 = y/x, z=sqrt5

cos(2a) = cos^2(a) - sin^2(a)

cos(a) = x/z, sin(a) = y/z

Answer 6

Where do you derive z=sqrt5 from? D:

Answer 7

but i cheated and used my fingers to "calculate" the z .... i got no idea how you would do it otherwise

Answer 8

trig function are right triangles ...

Answer 9

since x and y are defined by the tan; that leaves the pythag for the missing side

Answer 10

2^2+1^2 = 5 , right?

Answer 11

I'm quite certain you're supposed to do it through trigonometric relations etc; they're hinting with the application of the relations for cos(2x)

Answer 12

i did do it thru trig relations ... just in a more basic trig relation process :)

Answer 13

I don't follow. :(

Answer 14

@Noliec
The other way round,
\[\Huge \cos2x=\frac{1-\tan^2x}{1+\tan^2x}\]

Answer 15

substitute the values :)

Answer 16

How do you derive this? :O

Answer 17

not real sure how we could develop the cos(2x) = cos^2-sin^2 identity without some more fundamental processes

Answer 18

Oh, I see what you did DLS! :D

Answer 19

Thanks both of you! :) @DLS @amistre64

Answer 20

easy to derive :) but you don't really need it just keep it in mind

Answer 21

Really beautiful problem.