C3 trig double angle and binomial expansion

I had a question in the c3 book, i had to write the following in its simplest form and with only one trig function, Here it is

sin^4\theta-2sin^2 \theta cos^2 \theta+cos^4 \theta

after finding the answer

cos^2 2 \theta

(just barely, i actualy very much doubted this answer because of the way the original function looked i.e the

-2sin^2\theta cos^2 \theta

but it turned out to be the right answer, doesn't this look like binomial exapnsion? the onl thing stoping it from looking exacly like binomial is the

-

being there instead of

+

. another reason i did not think my answer was right was because

cos 2 \theta =cos^2 \theta-sin^ \theta

not the

+

that i had in the original function.

(edited 13 years ago)

Reply 1

nuodai

17

Your answer's right. Like, what do you get when you square

(u-v)^2

? So set

u=\cos \theta, v=\sin \theta

and see what happens.

Reply 2

Core

OP

6

cos ^2 \theta - 2sin\theta cos\theta +sin^2 \theta

thats one to remember, only thing is ive forgotten how i go the answer i think i tried to sqaure

cos^2 \theta -sin^ \theta

to get it into the form of the original function but i kept skipping the working as i thought it was going to be wrong, so i made assumption about it.

(edited 13 years ago)

Reply 3

nuodai

17

Original post by Core

cos ^2 \theta - 2sin\theta cos\theta +sin^2 \theta

Sorry, I meant

u=\cos^2 \theta

and

v=\sin^2 \theta

; my bad.

Reply 4

Core

OP

6

Original post by nuodai

Sorry, I meant

u=\cos^2 \theta

and

v=\sin^2 \theta

; my bad.

No worries you still helped, It seems as if there are alot of things im ging to have to "look out for" if i want to be good at solving this problems, such a as the one in this question, fortunately with practice and help ive been getting better.

Reply 5

nuodai

17

Original post by Core

No worries you still helped, It seems as if there are alot of things im ging to have to "look out for" if i want to be good at solving this problems, such a as the one in this question, fortunately with practice and help ive been getting better.

I think with this question it's just a case of noticing that you can factorise it. Usually the actual exam questions on trig identities are more obvious than this one, but it doesn't hurt to practice. There aren't many new tricks they can throw at you to be honest, so just practice questions like this and you'll have no problems.

Reply 6

Meeriee.me

can somene help mee Given that:

cos(x+30) = 3 cos(x-30) , prove that,

tan x = - ''root 3''/2

then prove that: (1 - cos '2theta') / sin '2theta' = tan 'theta'

and verify that 'theta' = 180 is a solution of the equation sin 2'theta' = 2 - 2 cos2'theta'

C3 trig double angle and binomial expansion

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