How do you write #2 +4i# in trigonometric form?

Question

Quinn Anderson · Accepted Answer

#2sqrt5(cos(1.107)+isin(1.107))# #"to convert to trigonometric form"# #"that is "x+yitor(costheta+isintheta)" using"# #•color(white)(x)r=sqrt(x^2+y^2)# #•color(white)(x)theta=tan^-1(y/x);-pi < theta <=pi# #"here "x=2" and "y=4# #rArrr=sqrt(2^2+4^2)=sqrt20=2sqrt5# #theta=tan^-1(2)=1.107" radians to 3 dec. places"# #rArr2+4i=2sqrt5(cos(1.107)+isin(1.107))#

Anna Andrews · Answer

undefined To write the complex number $2 + 4i$ in trigonometric form, we need to express it in terms of its magnitude and argument.

First, we find the magnitude ($r$) of the complex number using the formula:
$$ r = \sqrt{a^2 + b^2} $$
where $a$ is the real part and $b$ is the imaginary part of the complex number.

Given $a = 2$ and $b = 4$, we have:
$$ r = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} $$

Next, we find the argument ($\theta$) of the complex number using the formula:
$$ \theta = \arctan\left(\frac{b}{a}\right) $$
where $a$ is the real part and $b$ is the imaginary part of the complex number.

Given $a = 2$ and $b = 4$, we have:
$$ \theta = \arctan\left(\frac{4}{2}\right) = \arctan(2) $$

Now, to express $\theta$ in radians:
$$ \theta \approx 1.107 \, \text{radians} $$

Therefore, the trigonometric form of the complex number $2 + 4i$ is:
$$ 2\sqrt{5} \left(\cos(1.107) + i \sin(1.107)\right) $$