How do you write the trigonometric form of $2 + 2 i$ $2+2i$ ?

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How do you write the trigonometric form of $2 + 2 i$ $2+2i$ ?

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Answer 1 · 2016-10-25T12:07:13

The trigonometric form is $2 \sqrt{2} (cos (\frac{π}{4}) + i sin (\frac{π}{4}))$ $2sqrt2(cos(pi/4)+isin(pi/4))$

Explanation:

Let $z = 2 + 2 i$ $z=2+2i$
To calculate the trigonomrtric version, we need to calculate the modulus of the complex number.

If $z = a + i b$ $z=a+ib$ then the modulus is $∣ ∣ z ∣ = \sqrt{a^{2} + b^{2}}$ $∣z∣=sqrt(a^2+b^2)$

So here $∣ ∣ z ∣ = \sqrt{2^{2} + 2^{2}} = 2 \sqrt{2}$ $∣z∣=sqrt(2^2+2^2)=2sqrt2$
Then $\frac{z}{∣ z ∣} = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}$ $z/(∣z∣)=1/sqrt2+(i)/sqrt2$

tHen we compare this to $z = r (cos θ + i sin θ)$ $z=r(costheta+isintheta)$

and we get $cos θ = \frac{1}{\sqrt{2}}$ $costheta=1/sqrt2$
and $sin θ = \frac{1}{\sqrt{2}}$ $sintheta=1/sqrt2$

so, $θ = \frac{π}{4}$ $theta=pi/4$
and the trigonometric form is $2 \sqrt{2} (cos (\frac{π}{4}) + i sin (\frac{π}{4})$ $2sqrt2(cos(pi/4)+isin(pi/4)$

And the exponential form is $z = 2 \sqrt{2} e^{i \frac{π}{4}}$ $z=2sqrt2e^(ipi/4)$

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How do you write the trigonometric form of $2 + 2 i$ $2+2i$ ?

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How do you write the trigonometric form of $2 + 2 i$ $2+2i$ ?