Question

If sin(x)-cos(x)=1/2,find sin(x)+cos(x)

Answer 1

@hartnn we need your help!

Answer 2

Is this the equation:\[sin\theta-cos\theta=\frac{1}{2}\]

Answer 3

yes

Answer 4

Do they give you \(\theta\)?

Answer 5

no

Answer 6

any idea bro?

Answer 7

I am trying, just can't figure it out...

Answer 8

same here

Answer 9

What could we do to change \(sin\theta-cos\theta\) to \(sin\theta+cos\theta\)?

Answer 10

well,one could use the \[\left( a-b \right)^{2 } \] identity

Answer 11

How?

Answer 12

Blai is correct

with
\((\sin x - \cos x )^2 \)
you will get the value if
2 sin x cos x

that you can use in
\((\sin x + \cos x )^2 \)
and just take the square root :)

Answer 13

remember that
\(\sin^2 x +\cos^2 x =1\)

Answer 14

nice !

Answer 15

(sinx-cosx)^2=(1/2)^2

Answer 16

Great Job!

Answer 17

yes chad, let blai try out further steps :)

Answer 18

sin^2x-2sinxcosx+cos^2x=1/4

Answer 19

sorry

Answer 20

no problem :)

Answer 21

go ahead blai the platform is yours

Answer 22

The Questions the Stage, the mathematicians merely investigators.

Answer 23

@BlaiRe, are you still here ;)

Answer 24

This is all fine if \(\sin(x) \ge \cos(x)\). What if it isn't?

It might be interesting to exploit these:

\(\sin(x) - \cos(x) = \sqrt{2}\sin(x-\pi/4)\)

\(\sin(x) + \cos(x) = \sqrt{2}\sin(x+\pi/4)\)

Answer 25

if sin x < cos x,
wouldn't sin x - cos x be negative ?

sin x-cos x = 1/2 implies sin x is greater than cos x by 1/2

Answer 26

True, but when squaring, you have introduced other solutions. Are you SURE you have eliminated them along the way?

Answer 27

I'm here now

Answer 28

OK,here's what I've got after careful analysis of the above replies
\[\left( \sin \theta -\cos \theta \right)^{2} =\frac{ 1 }{ 4 } \]
\[1-2\sin \theta \cos \theta =\frac{ 1 }{ 4 }\]
\[-2\sin \theta \cos \theta =\frac{ 1 }{ 4 } -1 \]
\[\frac{ 1-4 }{ 4 }\]
\[\frac{ -3 }{ 4 } \]
\[2\sin \theta \cos \theta =\frac{ 3 }{ 4} \]
\[\left( \sin \theta +\cos \theta \right)^{2} = \sin ^{2} +\cos ^{2} +2\sin \theta \cos \theta \]
\[1+\frac{ 3 }{ 4 }\]
\[\frac{ 7 }{ 4 }\]
\[\sin \theta +\cos \theta =\frac{ \sqrt{7} }{ 2 }\]

Answer 29

thats correct :)
you would have got the same answer if sin x-cos x was -1/2

Answer 30

That's not quite correct. There are infinitely many solutions.