Question

The principle amplitude of ${(\sin {40^ \circ } + i\cos {40^ \circ })^5}$is:
A. ${70^ \circ }$
B. $- {1100^ \circ }$
C. ${70^{110}}$
D. ${70^{ - 70}}$

Answer 1

De Moivre’s Theorem: For any complex number z = {\left\\{ {r(\cos \theta + i\sin \theta )} \right\\}^n}, where n is any integer , the given complex number can be written as:
z = {\left\\{ {r(\cos \theta + i\sin \theta )} \right\\}^n} = {r^n}\left\\{ {\cos (n\theta ) + i\sin (n\theta )} \right\\}
Principal argument of any complex number$z = x + iy$is given as $\theta = {\tan ^{ - 1}}\dfrac{y}{x}$.

Complete step by step solution:
Given complex numbers ${(\sin {40^ \circ } + i\cos {40^ \circ })^5}$.
Simplifying the complex number:

\Rightarrow {(\sin {40^ \circ } + i\cos {40^ \circ })^5} \\\ \Rightarrow {\left\\{ {\sin {{(90 - 50)}^ \circ } + i\cos {{(90 - 50)}^ \circ }} \right\\}^5} \\\ \Rightarrow {\left\\{ {\cos {{50}^ \circ } + i\sin {{50}^ \circ }} \right\\}^5} \\\

$DeMoivre'sTheorem:$For any complex number z = {\left\\{ {r(\cos \theta + i\sin \theta )} \right\\}^n}, where n is any integer , the given complex number can be written as:
z = {\left\\{ {r(\cos \theta + i\sin \theta )} \right\\}^n} = {r^n}\left\\{ {\cos (n\theta ) + i\sin (n\theta )} \right\\}

$$\Rightarrow {(\cos {50^ \circ } + i\sin {50^ \circ })^5} \\\ \Rightarrow \left( {\cos (5){{\left( {50} \right)}^ \circ } + i\sin (5)\left( {50} \right)} \right) \\\ \Rightarrow (\cos {250^ \circ } + i\sin {250^ \circ }) \\\ \Rightarrow \cos {\left( {360 - 110} \right)^ \circ } + i\sin {\left( {360 - 110} \right)^ \circ } \\\ \Rightarrow \cos {110^ \circ } - i\sin {110^ \circ } \\\$$

Principal argument:

$$= {\tan ^{ - 1}}\left( { - \dfrac{{\sin {{110}^ \circ }}}{{\cos {{110}^ \circ }}}} \right) \\\ = {\tan ^{ - 1}}\left( { - \tan {{110}^{^ \circ }}} \right) \\\ = - {110^{^ \circ }} \\\$$

None of the given options is correct.
Correct answer is ${110^ \circ }$.

Note:
Principal Amplitude: The value of t which lies in the interval$( - \pi ,\pi )$, is called the principal amplitude of the complex number $z = x + iy$. Principal amplitude of a complex number can be found as follows.
$\Rightarrow t = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$
Consider the following points:
(a) principal argument will be $t$, if both a and b are positive.
(b) principal argument will be $\pi - t$, if both a and b are positive.
(c) principal argument will be $- \left( {\pi - t} \right)$, if both a and b are positive.
(d) principal argument will be $- t$, if both a and b are positive.