Expert: Paul Klarreich - 12/13/2009

Question
QUESTION: Q#1 Directions: Solve the equation exactly over the interval [0,360)
sin ^2 θ - cos ^2 θ = 0

Q#2 Directions solve the equation exactly over the interval [0,2π)
tan 2x-tan x = 0

#3 same directions as above,
2√3 sin 4x =3
for this problem my answers were π/4 and 5π/4 it was wrong
there 8 answers for this problem....

last problem,
directions solve the equation exactly
arcsin 2x + 2 arccos x =π
the answer is 0 but i have no idea how to do this

ANSWER: Questioner: dillon
Country: United States
Category: Advanced Math
Private: No
Subject: trig
Question: Q#1 Directions: Solve the equation exactly over the interval [0,360)
sin ^2 θ - cos ^2 θ = 0

Q#2 Directions solve the equation exactly over the interval [0,2π)
tan 2x-tan x = 0

#3 same directions as above,
2√3 sin 4x =3
for this problem my answers were π/4 and 5π/4 it was wrong
there 8 answers for this problem....

last problem,
directions solve the equation exactly
arcsin 2x + 2 arccos x =π
the answer is 0 but i have no idea how to do this
-------------------------------------------------------
#1:

Write   sin ^2 θ - cos ^2 θ = 0

0 = - sin ^2 θ + cos ^2 θ

cos (2t) = 0    <<< I can't make your theta;  't' will be used.

Now Cos x = 0 for x = 90, 270, and  450, 630.

You need this because if  t is in (0,360), then 2t is in (0,720) -- an important principle that you will use in #3 as well.

So 2t = 90, 270, 450, 630

etc.

..........................
#2:

tan 2x - tan x = 0

tan 2x = tan x

2 tan x
----------- = tan x
1 - tan^2 x


2 tan x  = tan x(1 - tan^2 x)

2 tan x  - tan x(1 - tan^2 x) = 0

2 tan x  - tan x + tan^3 x = 0

tan x  + tan^3 x = 0

tan x( 1 + tan^2 x) = 0

tan x = 0 has  x = 0, 180, as solutions.


1 + tan^2 x = 0 has no solutions.

Interesting.

.................................
#3

2√3 sin 4x = 3

sin 4x = 3/2 sqrt(3) = sqrt(3)/2

Now  4x = 60, 120, plus 360's as before.

Now divide by 4:

x = 15, 30, and six more.

..................................
#4

arcsin 2x + 2 arccos x =π

Whenever you have an arc...  remember that it is an angle:

Let  Y = arcsin 2x,  then 2x = sin Y

Let Z = arccos x,  then  x = cos Z

So arcsin 2x + 2 arccos x =π

becomes:

Y + 2Z = pi.

Y = pi - 2Z

sin Y = sin (pi - 2Z)

sin Y = sin pi cos(2Z) - cos pi sin(2Z)

sin Y =  sin(2Z)

sin Y =  2 sin Z cos Z

2x = 2 sin Z  x

x = x sin Z

-->  x = 0  or  sin Z = 1.

sin Z = 1  -->  Z = pi/2

--> x = cos Z = cos pi/2,  also  = 0.

So  x = 0 appears to be IT.

....................................

Trig is fun, WHEN YOU KNOW YOUR IDENTITIES.  Get back to work learning them.


---------- FOLLOW-UP ----------

QUESTION: trig is fun haha, that's a good one.
I am more of an algebra person myself, I think factoring equations is fun like x^2 -9 stuff like that. but not when the X is sine or tangent.

For the 2nd question, I did know that it didnt have a solution, I was doing some corrections on a test that I did, got it wrong, and I had to show all the steps to get it answer.

I didnt really understand the 4th question

And then has nothing to do with Trig, but i was looking at questions that you have been asked before, Science of Intelligent Universe, I didnt understand much of (vocabulary of a 16 year old...) but i thought it was interesting how the whole universe could be connected somehow. I was just wondering, is it relevant to compare the human brain to the universe?  

Answer
If you don't like factoring  sin^2(x), write S for sin x, as I often do.

Yes, No.4 was a bit involved.   Given some time, I could probably streamline the work a bit.

As to: "I was just wondering, is it relevant to compare the human brain to the universe?"

Sure, go ahead, have fun, just don't call it science.  Those journals the author is referring to are not true science journals.