Respuesta :
Given z = 3 + i, right away we can find
(a) square
z ² = (3 + i )² = 3² + 6i + i ² = 9 + 6i - 1 = 8 + 6i
(b) modulus
|z| = √(3² + 1²) = √(9 + 1) = √10
(d) polar form
First find the argument:
arg(z) = arctan(1/3)
Then
z = |z| exp(i arg(z))
z = √10 exp(i arctan(1/3))
or
z = √10 (cos(arctan(1/3)) + i sin(arctan(1/3))
(c) square root
Any complex number has 2 square roots. Using the polar form from part (d), we have
√z = √(√10) exp(i arctan(1/3) / 2)
and
√z = √(√10) exp(i (arctan(1/3) + 2π) / 2)
Then in standard rectangular form, we have
[tex]\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right)\right)[/tex]
and
[tex]\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right)\right)[/tex]
We can simplify this further. We know that z lies in the first quadrant, so
0 < arg(z) = arctan(1/3) < π/2
which means
0 < 1/2 arctan(1/3) < π/4
Then both cos(1/2 arctan(1/3)) and sin(1/2 arctan(1/3)) are positive. Using the half-angle identity, we then have
[tex]\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}[/tex]
[tex]\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}[/tex]
and since cos(x + π) = -cos(x) and sin(x + π) = -sin(x),
[tex]\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}[/tex]
[tex]\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1-\cos\left(\arctan\left(\dfrac13\right)\right)}2}[/tex]
Now, arctan(1/3) is an angle y such that tan(y) = 1/3. In a right triangle satisfying this relation, we would see that cos(y) = 3/√10 and sin(y) = 1/√10. Then
[tex]\cos\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1+\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10+3\sqrt{10}}{20}}[/tex]
[tex]\sin\left(\dfrac12 \arctan\left(\dfrac13\right)\right) = \sqrt{\dfrac{1-\dfrac3{\sqrt{10}}}2} = \sqrt{\dfrac{10-3\sqrt{10}}{20}}[/tex]
[tex]\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}[/tex]
[tex]\sin\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{10-3\sqrt{10}}{20}}[/tex]
So the two square roots of z are
[tex]\boxed{\sqrt z = \sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}[/tex]
and
[tex]\boxed{\sqrt z = -\sqrt[4]{10} \left(\sqrt{\dfrac{10+3\sqrt{10}}{20}} + i \sqrt{\dfrac{10-3\sqrt{10}}{20}}\right)}[/tex]
Answer:
[tex]\displaystyle \text{a. }8+6i\\\\\text{b. }\sqrt{10}\\\\\text{c. }\\\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}}+i\sqrt{\frac{\sqrt{10}-3}{2}},\\-\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}}-i\sqrt{\frac{\sqrt{10}-3}{2}}\\\\\\\text{d. }\\\text{Exact: }z=\sqrt{10}\left(\cos\left(\arctan\left(\frac{1}{3}\right)\right), i\sin\left(\arctan\left(\frac{1}{3}\right)\right)\right),\\\text{Approximated: }z=3.16(\cos(18.4^{\circ}),i\sin(18.4^{\circ}))[/tex]
Step-by-step explanation:
Recall that [tex]i=\sqrt{-1}[/tex]
Part A:
We are just squaring a binomial, so the FOIL method works great. Also, recall that [tex](a+b)^2=a^2+2ab+b^2[/tex].
[tex]z^2=(3+i)^2,\\z^2=3^2+2(3i)+i^2,\\z^2=9+6i-1,\\z^2=\boxed{8+6i}[/tex]
Part B:
The magnitude, or modulus, of some complex number [tex]a+bi[/tex] is given by [tex]\sqrt{a^2+b^2}[/tex].
In [tex]3+i[/tex], assign values:
- [tex]a=3[/tex]
- [tex]b=1[/tex]
[tex]|z|=\sqrt{3^2+1^2},\\|z|=\sqrt{9+1},\\|z|=\sqrt{10}[/tex]
Part C:
In Part A, notice that when we square a complex number in the form [tex]a+bi[/tex], our answer is still a complex number in the form
We have:
[tex](c+di)^2=a+bi[/tex]
Expanding, we get:
[tex]c^2+2cdi+(di)^2=a+bi,\\c^2+2cdi+d^2(-1)=a+bi,\\c^2-d^2+2cdi=a+bi[/tex]
This is still in the exact same form as [tex]a+bi[/tex] where:
- [tex]c^2-d^2[/tex] corresponds with [tex]a[/tex]
- [tex]2cd[/tex] corresponds with [tex]b[/tex]
Thus, we have the following system of equations:
[tex]\begin{cases}c^2-d^2=3,\\2cd=1\end{cases}[/tex]
Divide the second equation by [tex]2d[/tex] to isolate [tex]c[/tex]:
[tex]2cd=1,\\\frac{2cd}{2d}=\frac{1}{2d},\\c=\frac{1}{2d}[/tex]
Substitute this into the first equation:
[tex]\left(\frac{1}{2d}\right)^2-d^2=3,\\\frac{1}{4d^2}-d^2=3,\\1-4d^4=12d^2,\\-4d^4-12d^2+1=0[/tex]
This is a quadratic disguise, let [tex]u=d^2[/tex] and solve like a normal quadratic.
Solving yields:
[tex]d=\pm i \sqrt{\frac{3+\sqrt{10}}{2}},\\d=\pm \sqrt{\frac{{\sqrt{10}-3}}{2}}[/tex]
We stipulate [tex]d\in \mathbb{R}[/tex] and therefore [tex]d=\pm i \sqrt{\frac{3+\sqrt{10}}{2}}[/tex] is extraneous.
Thus, we have the following cases:
[tex]\begin{cases}c^2-\left(\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2=3\\c^2-\left(-\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2=3\end{cases}\\[/tex]
Notice that [tex]\left(\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2=\left(-\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2[/tex]. However, since [tex]2cd=1[/tex], two solutions will be extraneous and we will have only two roots.
Solving, we have:
[tex]\begin{cases}c^2-\left(\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2=3 \\c^2-\left(-\sqrt{\frac{\sqrt{10}-3}{2}}\right)^2=3\end{cases}\\\\c^2-\sqrt{\frac{5}{2}}+\frac{3}{2}=3,\\c=\pm \sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}[/tex]
Given the conditions [tex]c\in \mathbb{R}, d\in \mathbb{R}, 2cd=1[/tex], the solutions to this system of equations are:
[tex]\left(\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}}, \sqrt{\frac{\sqrt{10}-3}{2}}\right),\\\left(-\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}},- \frac{\sqrt{10}-3}{2}}\right)[/tex]
Therefore, the square roots of [tex]z=3+i[/tex] are:
[tex]\sqrt{z}=\boxed{\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}}+i\sqrt{\frac{\sqrt{10}-3}{2}} },\\\sqrt{z}=\boxed{-\sqrt{\sqrt{\frac{5}{2}}+\frac{3}{2}}-i\sqrt{\frac{\sqrt{10}-3}{2}}}[/tex]
Part D:
The polar form of some complex number [tex]a+bi[/tex] is given by [tex]z=r(\cos \theta+\sin \theta)i[/tex], where [tex]r[/tex] is the modulus of the complex number (as we found in Part B), and [tex]\theta=\arctan(\frac{b}{a})[/tex] (derive from right triangle in a complex plane).
We already found the value of the modulus/magnitude in Part B to be [tex]r=\sqrt{10}[/tex].
The angular polar coordinate [tex]\theta[/tex] is given by [tex]\theta=\arctan(\frac{b}{a})[/tex] and thus is:
[tex]\theta=\arctan(\frac{1}{3}),\\\theta=18.43494882\approx 18.4^{\circ}[/tex]
Therefore, the polar form of [tex]z[/tex] is:
[tex]\displaystyle \text{Exact: }z=\sqrt{10}\left(\cos\left(\arctan\left(\frac{1}{3}\right)\right), i\sin\left(\arctan\left(\frac{1}{3}\right)\right)\right),\\\text{Approximated: }z=3.16(\cos(18.4^{\circ}),i\sin(18.4^{\circ}))[/tex]
