alleryIf $z = \cos\theta + i\sin\theta$, find an expression for $z^n + \dfrac{1}{z^n}$.
A$\sin 2n\theta$
B$\cos n\theta$
C$2\cos n\theta$
D$2i\sin n\theta$
Answer & Solution
Correct answer: C. $2\cos n\theta$
**Apply De Moivre.** $z^n = \cos n\theta + i\sin n\theta$.
**Compute the reciprocal.** Since $|z| = 1$, $\dfrac{1}{z^n} = \overline{z^n} = \cos n\theta - i\sin n\theta$ (conjugate of a unit-modulus complex number is its inverse).
**Sum.** $z^n + \dfrac{1}{z^n} = 2\cos n\theta$ — the imaginary parts cancel.
**Symmetric companion.** Subtracting instead gives $z^n - \dfrac{1}{z^n} = 2i\sin n\theta$ (option B) — useful for deriving multiple-angle formulae, but it's not what the question asks for.
Keep practising Unit 8A: Areas Between Curves
Browse all Unit 8A: Areas Between Curves questions for AP Calculus AB, or explore more Mathematics topics.
Related questions
Let $\mathbf{A}=\langle 1,2,3\rangle$, $\mathbf{B}=\langle 2,0,1\rangle$, and $\mathbf{C}=For nonzero vector $\mathbf{A}$, which expression gives a vector in the same direction as When subtracting vectors, $\vec A-\vec B$ is evaluated asThe projection of vector $\vec A$ on vector $\vec B$ is given byWhich statement about a null vector is correct?If $\omega$ is a non-real cube root of unity, find the value of $1 + \omega + \omega^2$.Find the modulus of the complex number $z = 3 + 4i$.The locus of the point $z$ in the Argand plane satisfying $|z - 2| = |z + 2|$ is: