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

Hint:
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.