Question

Will Medal!

If 2tanx/ (1-tan^2x) = 1, then x can equal ____.
Check all that apply.

A) x=5pi/8 + npi
B) x=pi/8 + npi
C) x=3pi/8 + npi
D) x=7pi/8 + npi

Answer 1

I think it is A and B. But this is me taking a guess. I am terrible at trig. Sorry

Answer 2

well first cross multiply to give
2 tanx = 1 - tan^2 x
now rearrange this quadratic:
tan^2 x + 2tanx - 1 = 0

can you factor this?

Answer 3

Yes. I am thinking you factor to get tan^2x = -2tan(x)+1?

Answer 4

no factor as you would any quadratic
like
x^2 + 3x + 2
= (x + 1)(x + 2)

Answer 5

oh sorry - on second look this will not factor
so you need to use the quadratic formula

Answer 6

So would a = tan^2x b =2tanx and c=-1?

Answer 7

no a , b and c are the coefficients of tan^2x and tan x
so a = 1 , b = 2 and c = -1

Answer 8

So x = +/- sqrt (2) in the end right?

Answer 9

*-1 +/- sqrt(2)

Answer 10

-2 +/- sqrt(4 - 4*1*-1) / 2
= -2 +/- sqrt8 / 2
= (-2 +/- 2 * sqrt2 )/ 2
= -1 +/- sqrt2


yep

Answer 11

Awesome! So now do I just plug in this to my original equation?

Answer 12

well they have the answer options in terms of pi
so convert everything to decimals

so for example -1 + sqrt2 = 0.4142
so find the angle in radians which is the tangent of 0.4142
and see if one of the options gives the the same angle

Answer 13

eg angle with tangent 0.4142 = 0.39269 radians
now check if pi/8 gives the same value

Answer 14

Yes it does!

Answer 15

So I'm assuming that is one

Answer 16

ok so thats one result
pi/8 + npi

Answer 17

now try the other value for tan x

Answer 18

-1 - sqrt 2 = -2.4142 degrees right?

Answer 19

yes

Answer 20

Ok so then the radians of -2.4142 is -0.042135

Answer 21

I don't think I did it right

Answer 22

no the -2.4142 is the value of tan x
so you need tan-1(-2.4242)

Answer 23

Oh ok that would make it -1.1781 which is basically the same as 3pi/8 right?

Answer 24

So the answers would be pi/8 + npi and 3pi/8 + npi

Answer 25

yes

Answer 26

Great!! Thank you for taking the time to help me! I feel I really understand now! Thank you!

Answer 27

Weird... I hit the sumit button and it told me it was 5pi/8 instead of 3/pi/8

Answer 28

lets check that out

Answer 29

\[\frac{2tanx}{1-\tan^2x} = 1 \\ \tan(2x)=1\]

Answer 30

\[\tan(u)=1 \text{ when } u=?\]

Answer 31

ahh
i forgot that identity !

Answer 32

u = pi/4 of course
and x will be pi/8

Answer 33

much easier that way lol!

Answer 34

and there is also the u=5pi/4
so yep x=5pi/8

Answer 35

yes

Answer 36

-1.1781 is an incorrect result then?
- an extraneous result right?

Answer 37

does give tan(2x) approx 1
what number are you approximating ?

Answer 38

oh -3pi/8 is what you are approximating

Answer 39

sorry I dont follow you
-yes

Answer 40

that gives 1

Answer 41

when you plug in

Answer 42

and -3pi/8 is equivalent to 5pi/8

Answer 43

is that not included in the solution:
\[\frac{5\pi}{8}+n \pi \\ \text{ if } n=-1 \text{ then } \frac{5\pi}{8}-\pi=\frac{-3\pi}{8}\]
so I'm not sure why you say it is extraneous

Answer 44

no - i was wrong its not extraneous

Answer 45

still i think lehmad learned something along the line...

Answer 46

Yes I learned a lot! Thank you guys for the help.