Question

What is the amplitude of the function \[\sin\dfrac{\pi }{5} + i\left( {1 - \cos\dfrac{\pi }{5}} \right)\]?
A. \[\dfrac{{2\pi }}{5}\]
B. \[\dfrac{\pi }{{15}}\]
C. \[\dfrac{\pi }{{10}}\]
D. \[\dfrac{\pi }{5}\]

Answer 1

Hint: Simplify the given function in the polar form of a complex number, by using the trigonometric identities. Then use the formula of the amplitude of a complex number \[\theta = \tan^{ - 1}\left( {\dfrac{y}{x}} \right)\], to reach the required answer.

Formula Used:
\[\sin2A = 2\sin A \cos A\]
\[2\sin^{2}A = 1 - \cos A\]
The amplitude of a complex number \[z = x + iy\] is: \[\theta = \tan^{ - 1}\left( {\dfrac{y}{x}} \right)\].

Complete step by step solution:
The given function is \[f\left( x \right) = \sin\dfrac{\pi }{5} + i\left( {1 - \cos\dfrac{\pi }{5}} \right)\].
Let’s simplify the above equation.
Apply the trigonometric identities \[2\sin^{2}A = 1 - \cos A\] and \[\sin2A = 2\sin A \cos A\].
\[f\left( x \right) = 2\sin\left( {\dfrac{\pi }{{10}}} \right)\cos\left( {\dfrac{\pi }{{10}}} \right) + i\left( {2\sin^{2}\left( {\dfrac{\pi }{{10}}} \right)} \right)\]
Factor out the common terms.
\[f\left( x \right) = 2\sin\left( {\dfrac{\pi }{{10}}} \right)\left[ {\cos\left( {\dfrac{\pi }{{10}}} \right) + i\sin\left( {\dfrac{\pi }{{10}}} \right)} \right]\]
The above function is in the polar form of a complex number.
Compare the above function with the polar form \[z = r\left( {\cos\theta + i\sin\theta } \right)\].
We get
\[r = 2\sin\left( {\dfrac{\pi }{{10}}} \right)\], \[x = 2\sin\left( {\dfrac{\pi }{{10}}} \right)\cos\left( {\dfrac{\pi }{{10}}} \right)\], and \[y = 2\sin^{2}\left( {\dfrac{\pi }{{10}}} \right)\]

Now apply the formula of amplitude \[\theta = \tan^{ - 1}\left( {\dfrac{y}{x}} \right)\].
\[\theta = \tan^{ - 1}\left( {\dfrac{{2\sin^{2}\left( {\dfrac{\pi }{{10}}} \right)}}{{2\sin\left( {\dfrac{\pi }{{10}}} \right)\cos\left( {\dfrac{\pi }{{10}}} \right)}}} \right)\]
\[ \Rightarrow \]\[\theta = \tan^{ - 1}\left( {\dfrac{{\sin\left( {\dfrac{\pi }{{10}}} \right)}}{{\cos\left( {\dfrac{\pi }{{10}}} \right)}}} \right)\]
\[ \Rightarrow \]\[\theta = \tan^{ - 1}\left( {\tan\left( {\dfrac{\pi }{{10}}} \right)} \right)\] [Since \[\dfrac{{\sin x}}{{\cos x}} = \tan x\]]
\[ \Rightarrow \]\[\theta = \dfrac{\pi }{{10}}\] [Since \[\tan^{ - 1}\left( {\tan A} \right) = A\]]

Hence the correct option is C.

Note: The polar representation of a complex number \[z = x + iy\] is \[z = r \left( {\cos \theta + i \sin \theta } \right)\].
Where, \[r\] is the modulus of the complex number and \[r = \sqrt {{x^2} + {y^2}} \].
\[\theta \] is the amplitude or argument of the complex number. It is denoted by \[Arg\left( z \right)\] or \[Amp\left( z \right)\].