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Home » Mathematics » High School Intermediate Topics (Ages 13+)
 
Trig Identity
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veezbo
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#1
Trig Identity
Sin x + Cos x
Prove that \sin x + \cos x = \sqrt{2}\sin(\theta + \pi/4)
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PostPosted: Sat Nov 07, 2009 8:12 pm  Back to top 
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ckck
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#2
Re: Trig Identity
Sin x + Cos x
veezbo wrote:
Prove that \sin x + \cos x = \sqrt {2}\sin(\theta + \pi/4)

Assuming you meant \sin\theta+\cos\theta Razz

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From the LHS factor out {\sqrt{2} to get \sqrt{2}(\frac{\sqrt{2}}{2}\sin\theta+\frac{\sqrt{2}}{2}\cos\theta)
\sqrt{2}(\cos\frac{\pi}{4}\sin\theta+\sin\frac{\pi}{4}\cos\theta)
\sqrt{2}\sin(\theta+\frac{\pi}{4})
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PostPosted: Sat Nov 07, 2009 8:23 pm  Back to top 
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abcak
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#3
Re: Trig Identity
Sin x + Cos x
veezbo wrote:
Prove that \sin x + \cos x = \sqrt {2}\sin(\theta + \pi/4)

I didn't get your proof ckck.

Don't you just replace sin(\theta+\pi/4) with cos(\theta), square both sides, and use a few identities?

PostPosted: Sun Nov 08, 2009 6:01 pm  Back to top 
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ckck
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#4
Re: Trig Identity
Sin x + Cos x
abcak wrote:
veezbo wrote:
Prove that \sin x + \cos x = \sqrt {2}\sin(\theta + \pi/4)

I didn't get your proof ckck.

Don't you just replace sin(\theta+\pi/4) with cos(\theta), square both sides, and use a few identities?

Uh what don't you get about my proof maybe
I just factored out \sqrt{2}, and used the fact that \frac{\sqrt{2}}{2}=\sin\frac{\pi}{4}=\cos\frac{\pi}{4} to simplify the expression into what we needed it to be.
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PostPosted: Sun Nov 08, 2009 6:10 pm  Back to top 
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adbh94
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#5
He's using the sum of angles formula between steps 2 and 3. That might be what is confusing.

PostPosted: Sun Nov 08, 2009 6:12 pm  Back to top 
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veezbo
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#6
Re: Trig Identity
Sin x + Cos x
abcak wrote:
veezbo wrote:
Prove that \sin x + \cos x = \sqrt {2}\sin(\theta + \pi/4)

I didn't get your proof ckck.

Don't you just replace sin(\theta+\pi/4) with cos(\theta), square both sides, and use a few identities?


This is what tripped me up when I first read this question. \pi/4 = 45 degrees not 90 degrees

So \sin(\theta+\pi/4) \neq \cos(\theta)

ckck's way of doing it is probably the easiest
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PostPosted: Sun Nov 08, 2009 6:29 pm  Back to top 
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grn_trtle
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#7
You could also expand out the right side.

\sqrt{2}\left(\sin\theta\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\cos\theta\right)=\sin\theta+\cos\theta.

This is a bit more obvious than factoring \sqrt{2} out of \sin\theta+\cos\theta.
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PostPosted: Sun Nov 08, 2009 10:48 pm  Back to top 
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Dr Sonnhard Graubner
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#8
hello, you can also write \sin(x)+\cos(x)=\sqrt{2}\cos(\frac{\pi}{4}-x)
Sonnhard.

PostPosted: Mon Nov 09, 2009 11:36 am  Back to top 
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