Write the expression in terms of sine only: -3cos2t+5sin2t

Question

owlcoffee · Answer

like this? 
$-3 \cos2t+5\sin2t$ ?
or
$-3\cos^2t+5\sin^2t$ ?

mtepecz · Answer

The first one

imqwerty · Answer

$$\cos2t=1-2\sin^{2}t$$
just substitute

jhannybean · Answer

Remember that $\cos(2t) = 1-2\sin^2(t)$

mtepecz · Answer

@imqwerty @Jhannybean where do we get the cos (2t) from again?

imqwerty · Answer

do u know this identity-$$\cos(A+B)=cosAcosB-sinAsinB$$
??

mtepecz · Answer

oh yeahhh

imqwerty · Answer

yea try to replace A and B with t 
what will u get?

mtepecz · Answer

2t?

imqwerty · Answer

yes
 and on the right hand side?

baru · Answer

i think the question is asking to write in the form  Asin$(2 t-\phi)$

mtepecz · Answer

yeah i have to write it with the sine formula

imqwerty · Answer

yes we will do that 
but step wise :)

jhannybean · Answer

Seems to me everything just has to be in terms of sine at the end. Why not replace $\cos(2t)$ in your problem with $1-\sin^2(t)$ andf see where that takes you?

imqwerty · Answer

@mtepecz  u stuck somewhere?

mtepecz · Answer

I'm just confused on why its $$\cos( 2t)$$ ansd not $$\cos ^{2}(2t)$$

imqwerty · Answer

u mean like when we put t in this-> cos(A+B)
then we get cos(2t) but y not cos^2(2t) 
right?

mtepecz · Answer

yeah

mtepecz · Answer

Yes

imqwerty · Answer

ok now lets convert that $$\cos^2(t) $$ into sine terms
can u write it in terms of sine??

mtepecz · Answer

$$\cos ^{2}t=1-\sin ^{2}t$$

imqwerty · Answer

yes correct :)
now we put this value of cos^2(t) in our equation
we get this-$$\cos(2t)=\cos^2(t)-\sin^2(t)$$now we put the value$$\cos(2t)=1-\sin^2(t)-\sin^2(t)$$simplifying$$\cos(2t)=1-2\sin^2(t)$$
ok?  :)

mtepecz · Answer

is that all or do we have to square root it?

imqwerty · Answer

no no 
now we have an identity cos(2t)=1-2sin^2 (t)
now u look ta ur question an see where u can use this identity

mtepecz · Answer

I honestly don't know

imqwerty · Answer

its ok :)
do u get that cos(2t)=1-2sin^2(t)   ?

mtepecz · Answer

Yeah I get that, but what do we do next? Plug in numbers?

imqwerty · Answer

ok now lets look at our original equation

imqwerty · Answer

we have this$$-3\cos(2t)+5\sin(2t)$$
we need to bring all the terms in sine

can u tell me which term is not in sine

mtepecz · Answer

Would it be the right side?

imqwerty · Answer

what do u mean by right side?

mtepecz · Answer

5sin 2t

imqwerty · Answer

no :)
well we need to bring all terms in sine form
but 5sin(2t) is already in sine form 

but -3cos(2t)  is in cosine form and we have to deal with it

ok?

mtepecz · Answer

okay

imqwerty · Answer

so we look at -3cos(2t)

we see that this thing has got cos(2t) in it!
and now we use our identity 
can u apply our identity in this?

ganeshie8 · Answer

Hey, as baru said, we could also write it as
$$-3\cos 2t + 5\sin 2t = \sqrt{3^2+5^2} \sin(2t-\phi)$$

mtepecz · Answer

Is there going to be a square root at some point? I feel like what I'm doing is wrong

mtepecz · Answer

@imqwerty

imqwerty · Answer

no ur not wrong
are u feeling like -on one side its a linear term cos(2t) and on the other side we have sin^2 (t) which is a squared term so we shuld square root
?

mtepecz · Answer

yes

imqwerty · Answer

haha don't worry then
u can try an example if u want 
for example u take t=45degrees :)
now u put it in our identity to check if it holds true

mtepecz · Answer

no

imqwerty · Answer

it didn't hold true?

mtepecz · Answer

no I probably did it wrong

imqwerty · Answer

check ur calculations once again

mtepecz · Answer

I'm sorry to have wasted your time, I'm tired and don't get it but thank you :)