Question

Solve algebraically for the value of x in the interval

0 (smaller or equal to) x (less than) 2pi

Answer 1

\[4sinxcosx = 2-2\cos^2x\]

Answer 2

@Hero

Answer 3

Why not simplify matters by dividing every term by 2? Not strictly necessary, but nice.

Answer 4

Next, look at the right side. Can you think of an identity for it?

Answer 5

yes sin^2x

Answer 6

that might be a possibility. Were it not for that coefficient 2 on the left side, I'd try rewriting the right side in the form (sin x)^2.

Your turn. Ideas?

Answer 7

so you mean divide evertything by 2 first?

Answer 8

Is there an identity for 2 sin x cos x?

Answer 9

I did suggest you divide every term of the original equation by 2, yes.

type out what you have now as a result.

Answer 10

ok a moment

Answer 11

2 sin x cos x = 1 - 2 cos^2 x

sin 2 x = sin^2 x

Answer 12

Yes, that's what I expected. You've eliminated cos x, which is good.
See any way of simplifying further?

Answer 13

we should not have done that

2 sinx cos x = sin ^2 x

sinx on both side could cancel out

2 = sin x / cos x

2 = tanx

Answer 14

I was thrown a bit by your sin^2x. I read that as "the sine of 2 times x."

Best to write "the square of sin x" as (sin x)^2 or \[\sin^2x\]

Answer 15

yes sorry

Answer 16

so am I right ?

Answer 17

Rather than check your work further, I'll suggest you look for solutions of tan x = 2.

Answer 18

Remember that tan x is defined for all x other than pi/2, pi+pi/2, and so on.

Answer 19

So you may find several solutions within the interva [0, 2pi].

Answer 20

Excuse me, [0, 2pi)

Answer 21

yes

tan x = 2

is positive in the first and third quadrant using the CAST rule
so tan ^ -1 (2) = 1.107

Answer 22

yes several solutions are present

Answer 23

One way to get a handle on this is to graph tan x on the interval [0,2p) and then draw a horiz. line thru y=2. How many times does this line hit the curves of tan x?

Answer 24

Recall that the period of tan x is pi (not 2pi).

Answer 25

yes

Answer 26

so it hit at 1.107 radian and 4.249 radian

Answer 27

and 0

Answer 28

It's important that you check your answers in the original equation. Set your calc to "RAD" mode and evaluate each side of the equ'n separately. Are the 2 sides =?

Answer 29

Sorry I don't understand what you mean here

Answer 30

It's important that you check your answers in the original equation. How would you do that?

You've shared your roots in radian measure, so if you check them in the original equation, the calculator has to be set to RAD mode first.

Answer 31

Are you Russian? I learned a semester's worth of Russian at UCLA in 1965. Seem to remember a word that sounds like khoroshoi, but can't remember its meaning.

Answer 32

I see what you mean, yes my calculator has to be in radian mode at all times

Answer 33

I am Persian :D

Answer 34

I know there are common names in languages

Answer 35

so there are 4 answers but I got 2

Answer 36

these are the four answers     x = 0 , pi , 1.107, 4.249

Answer 37

We might have to go back to the orignal equation and double-check the substitutions we made.
It's promising that you've obtained two that are correct.

I believe you cancelled sin x or cos x.
Note that if you actually did that, you may have lost a source of additional solutions inadvertently.

Persian, as in Iranian?

Answer 38

yeah I canceled something out. I should not have done that.


Yes Persian and Iranian are the same thing. Hey, you know that :D

Answer 39

note that sin x=0 has roots 0 and pi within the interval in question,
and that cos x=0 has the roots pi/2 and 3pi/2

Answer 40

Are you currently in the USA, or in Iran?

Answer 41

Canada

Answer 42

Neat. Fervently wish that the US and Iran could improve their relationship.

Answer 43

Could I possibly be of any further help with this particular trig problem?

Answer 44

Yeah, there I hope too

Answer 45

well give me a sec I will tell you

Answer 46

when we have

2 sin x cos x = 1- cos ^2 x

2 sinx cos x = sin ^2 x

what should I do after here?

Answer 47

Try factoring out sin x from both sides, but do not discard that factor.

What's left?

Answer 48

Or, rather, what are the resulting factors?

Answer 49

sinx (2cosx) = sinx ( six)

Answer 50

sin x (2 cos x ) = sin x (sin x)

Answer 51

yes, or (sin x)(2cos x-sin x)=0

that's not too promising, though.

Answer 52

I believe yes we got the answer

Answer 53

2 sin x cos x = sin^2 x

sin^2x - 2 sinx cos x= 0
sinx ( sinx - 2 cos x) = 0

sin x = 0 and
sin x = 2 cos x which is 2 = tan x

so sin ^ -1 (0 ) = 0 and pi?

Answer 54

What you've done looks good, correct.

"the angle whose sine is 0" is 0, but also pi.

The inverse sine of x at x=0 is {0, pi}.

Answer 55

Fantastic, nice we solved together.
Sorry if it took long time.

Thanks for backup :D

Answer 56

Very happy to have this opp to work with you. Later, perhaps? Take care.

Answer 57

sure we should solve more questions in the future, you too