Question

find the exact value of cos(2 arctan 4/3) without a calculator? please answer with details and explanations if you can.. i'm really confused and there's a test tomorrow. :(

Answer 1

\[\cos(2\theta)=2\cos^2(\theta)-1\] is a start

Answer 2

in this case \(\theta)=\arctan(\frac{4}{3})\)

Answer 3

so your job is first to find \(\cos(\arctan(\frac{4}{3}))\)
do you know how to do that?

Answer 4

it is the same as the question
" if the tangent of \(\theta\) is \(\frac{4}{3}\) find the cosine of \(\theta\) "

Answer 5

any ideas?

Answer 6

to find cos (2 archtan 4/3) i draw tan 4/3 and then find the cos on the triangle right? so cos would be 3/5?

Answer 7

yes!! exactly

Answer 8

and since \(\cos(\theta)=\frac{3}{5}\) then you know
\[\cos(2\theta)=2\left(\frac{3}{5}\right)^2-1\]

Answer 9

or maybe i should say
" since you know \(\cos(\tan^{-1}(\frac{3}{4}))=\frac{3}{5}\) you know \[\cos(2\tan^{-1}(\frac{3}{5}))=2\left(\frac{3}{5}\right)^2-1\]

Answer 10

okay thanks! but does it matter which cos(2θ) formula i use? any of the cos(2θ) formulas would work for this question right?

Answer 11

typo there, i meant
\[\cos(2\tan^{-1}(\frac{3}{4}))=2\left(\frac{3}{5}\right)^2-1\]

Answer 12

well lets think for a second
the formula
\[\cos(2\theta)=2\cos^2(\theta)-1\] contains only cosine
if you use the one with sine, you will have to find the sine

Answer 13

so the answer is "yes" you can use whichever one you like, but you want to make life easy, not hard

Answer 14

oh! that makes a lot of sense :)
so technically, i could use any (2θ) formula as long as i find the necessary trig function (sin/cos/tan) for it?

Answer 15

okay thank you so much!! :D

Answer 16

yw

Answer 17

@satellite73 i have a small question.. if you're not too busy, could you answer this really quick please?
-- tan(2 arcsin (-3/5)) --
would i solve this the same way as my original question? like find tan(2 arcsin (-3/5)) first, then use tan(2θ) to solve it?

Answer 18

nvm, i got it! :D