Consider the complex number $z = 9\angle 170^\circ$. What is the smallest positive integer $n$ such that $z^n$ is a real number? $n =$ Number
Added by Michelle H.
Close
Step 1
We want to find the smallest positive integer $n$ such that $z^n$ is a real number. We can write $z$ in polar form as $z = 9(\cos(170^\circ) + i\sin(170^\circ))$. Then $z^n = 9^n(\cos(170^\circ n) + i\sin(170^\circ n))$. For $z^n$ to be a real number, the Show more…
Show all steps
Your feedback will help us improve your experience
Babita Kumari and 92 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
In Exercises $21-40$, find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\sqrt{3}(\arctan (-\sqrt{2})) $$
Applications of Trigonometry
Polar Form of Complex Numbers
A sequence of complex numbers is defined by z_0 = 1 + sqrt{3}i, z_1 = -1/sqrt{2} + sqrt{3}/sqrt{2}i and z_n = (cos(pi/4) + sin(pi/4)i) z_{n-1} / |z_{n-2}| for n in N with n >= 2. (a) Write down the complex numbers z_0 and z_1 in polar form. No justification is required. (b) Determine with justification the smallest positive integer p such that, for every positive integer n, |z_{n+p}| = |z_n|. (c) Determine with justification the smallest positive integer q such that, for every positive integer n, z_{n+q} = z_n.
Sam S.
Consider the equation z^11 = (-1 + i). Find the value of z which satisfies this equation and which has the second smallest positive argument theta, 0 < theta < 2pi. Express your answer as z = re^(itheta), where r = and theta =. When entering your answers use "pi" as needed, and for roots use a^(1/n) for the nth root of the number a (e.g. 2^(1/5) is how we enter the 5th root of 2).
Sri K.
Recommended Textbooks
Elementary and Intermediate Algebra
Algebra and Trigonometry
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD