# How to add cosines with different phases using phasors

So, here's a question:

$\cos( \omega t ) + 2 \cos( \omega t + \frac{\pi}{4} ) + 3 \cos( \omega t + \frac{\pi}{2} )$

To add these together, I figure there should be at least 2 ways:

1) Cosine addition laws:

$$\cos( \omega t ) + 2 \left( \cos( \omega t ) \cos( \frac{\pi}{4} ) - \sin( \omega t ) \sin( \frac{\pi}{4} ) \right) + 3 \left( \cos( \omega t ) \cos( \frac{\pi}{2} ) - \sin( \omega t ) \sin( \frac{\pi}{2} ) \right)$$

$$=\cos( \omega t ) \left( 1 + \sqrt{2} \right) - \sin( \omega t ) \left( 3 + \sqrt{2} \right)$$

2) Phasors / complex addition

$$1 \angle 0 + 2 \angle 45 ^\circ + 3 \angle 90^\circ$$

$$= 1 + \sqrt{2} + j \sqrt{2} + 3 i$$

$$= 1 + \sqrt{2} + j ( 3 + \sqrt{2} )$$

Which has

$A = \sqrt{ 14 + 8 \sqrt{2} } \approx 5.03$

$\phi = \arctan{ \left( \frac{ 3 + \sqrt{2} }{ 1 + \sqrt{2} } \right) } \approx 1.07 rad \approx 61 ^\circ$

Thus answer is $5 \angle 61^\circ$, or $5 \cos( \omega t + 1.07 )$

If you graph them, $5 \cos( \omega t + 1.07 )$ produces the same graph as $\cos( \omega t ) \left( 1 + \sqrt{2} \right) - \sin( \omega t ) \left( 3 + \sqrt{2} \right)$

How can you convert between them?

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## 1 Answer

You have $$\tan\phi = \frac{\sin\phi}{\cos\phi}= \frac{3+\sqrt{2}}{1+\sqrt{2}}$$ Since $0<\phi<\pi/2$ we know that $\sin\phi,\cos\phi>0$. Therefore, $\sin\phi$ and $\cos\phi$ are equal to $$\sin\phi = \frac{3+\sqrt{2}}{\sqrt{(3+\sqrt{2})^2+(1+\sqrt{2})^2}} =\frac{3+\sqrt{2}}{\sqrt{14+8\sqrt{2}}}$$ and $$\cos\phi = \frac{1+\sqrt{2}}{\sqrt{(3+\sqrt{2})^2+(1+\sqrt{2})^2}} =\frac{1+\sqrt{2}}{\sqrt{14+8\sqrt{2}}}$$ Therefore $$\begin{split} \cos(\omega t)(1+\sqrt{2}) - \sin(\omega t)(3+\sqrt{2}) &= \sqrt{14+8\sqrt{2}}\left( \cos(\omega t)\cos\phi - \sin(\omega t) \sin\phi \right)\\& = \sqrt{14+8\sqrt{2}} \cos(\omega t+\phi) \end{split}$$

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$\sqrt{14+8\sqrt{2}}$, not $\sqrt{14+\sqrt2}$ – pharmine Nov 5 '11 at 9:03
@pharmine you're right, I've fixed it. – Heike Nov 5 '11 at 9:56