Question

tan ^{2}theta+(1-sqrt{3})tan theta=sqrt{3}

[0º≤θ<360º]

Find Θ.

Answer 1

\[\tan^{2}\theta+(1-\sqrt{3})\tan \theta=\sqrt{3}\]

Answer 2

Notice it looks just a little like a quadratic equation... subtract sqrt3 from both sides

\[\large \tan^{2}\theta+(1-\sqrt{3})\tan \theta - \sqrt{3} = 0\] \[\large (\tan\theta)^2+(1-\sqrt{3})\tan \theta - \sqrt{3} = 0\]
Now to make it more obvious, let \[x=\tan \theta\]
\[\large x^2 + (1-\sqrt{3})x - \sqrt3 = 0\]

Keep in mind that 1-sqrt3 is really just a constant, so we now have ax^2+bx+c=0.

Answer 3

does it have any other method except this?

Answer 4

how about common factors?

Answer 5

Well, you can factor it now that it's a quadratic.

Answer 6

\[x=-1/\sqrt{3}\]

Answer 7

θ=60/-45(rej.)

Answer 8

\[\large x^2 + (1-\sqrt{3})x - \sqrt3 = 0\] you can do it this way by distributing:
\[\large x^2 + x-\sqrt{3} x - \sqrt3 = 0 \]

then factor:
\[\large x(x + 1)-\sqrt{3}(x - 1) = 0 \]
\[\large (x-\sqrt{3})(x + 1) = 0\]
Now you can solve for x, and then use x = tan(theta)

Answer 9

Oops, 2nd to last line should be x+1 not x-1.

Answer 10

i only got 60 degrees

Answer 11

x=sqrt3 and x=-1, and x=tan(theta). There's... looks like 4 solutions total for 0º≤θ<360º

\[\tan \theta = \sqrt 3\]
\[\tan \theta = -1\]

Find them on the unit circle, http://jwilson.coe.uga.edu/emat6680fa06/crumley/unit/6_files/image010.jpg

Answer 12

yup, there have 4 answers

Answer 13

Did you find all 4?

Answer 14

Just look for where tan = sqrt 3 and -1.

Answer 15

nope, i just find 60degrees

Answer 16

Look harder :P

Answer 17

I can see four instances on here: http://jwilson.coe.uga.edu/emat6680fa06/crumley/unit/6_files/image010.jpg

where tan is either equal to -1, or sqrt3.

Answer 18

Here's two:

Answer 19

ummm...

Answer 20

180+(-45)=135

Answer 21

Do you know how to use a unit circle? all the values you need are on it.

Answer 22

no..

Answer 23

http://withfriendship.com/images/b/8803/Unit-circle-picture.gif

It shows the angles, and the values of cosine and sine.

Answer 24

tan (x + 180°) = tan x

you can also use this to find all of your solutions.

Answer 25

180+(-45)=135

with this 135 and 60 degrees, you can find them all.