Question

The amplitude of \[\sin \dfrac{\pi }{5} + i\left( {1 - \cos \dfrac{\pi }{5}} \right)\] is…

Answer 1

Hint: We are given a complex number form of trigonometric function commonly known as Euler's formula that gives the relation between trigonometric functions and the complex exponential function. We will use this relation to find the amplitude of the given function.

Complete step by step answer:
Given is the expression,
\[\sin \dfrac{\pi }{5} + i\left( {1 - \cos \dfrac{\pi }{5}} \right)\]
We know that, \[1 - \cos 2\theta = 2{\sin ^2}\theta \] and \[\sin 2\theta = 2\sin \theta \cos \theta \]
Using these we can write,
\[z = 2\sin \dfrac{\pi }{{10}}\cos \dfrac{\pi }{{10}} + i2{\sin ^2}\dfrac{\pi }{{10}}\]
Now taking the sin function along with 2 common,
\[z = 2\sin \dfrac{\pi }{{10}}\left( {\cos \dfrac{\pi }{{10}} + i\sin \dfrac{\pi }{{10}}} \right)\]
As we mentioned above Euler’s formula \[\cos \theta + i\sin \theta = {e^{i\theta }}\] gives the relation between trigonometric functions and the complex exponential function
Thus, the bracket can be written as,
 \[z = 2\sin \dfrac{\pi }{{10}}\left( {{e^{i\dfrac{\pi }{{10}}}}} \right)\]
We know that amplitude is given by,
\[\left| z \right| = \left| {2\sin \dfrac{\pi }{{10}}\left( {{e^{i\dfrac{\pi }{{10}}}}} \right)} \right|\]
But, \[\left| {{e^{i\theta }}} \right| = 1.....\forall \theta \]
Thus we can write,
\[\left| z \right| = 2\sin \dfrac{\pi }{{10}}\]
We know that the value of sin angle is,
\[\left| z \right| = 2\left( {\dfrac{{\sqrt 5 - 1}}{4}} \right)\]
Cancelling 4 with 2,
\[\left| z \right| = \dfrac{{\sqrt 5 - 1}}{2}\]
This is the amplitude of the function above.
Therefore, the amplitude of \[\sin \dfrac{\pi }{5} + i\left( {1 - \cos \dfrac{\pi }{5}} \right)\] is \[\dfrac{{\sqrt 5 - 1}}{2}\].

Note:
Note that, $z$ is the significance of the number to be complex. So we used that letter. The amplitude for such function is given by the modulus sign and for purely complex numbers of the form x+iy it is given by, \[\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)\] such that \[r = \cos \theta + i\sin \theta \] is the polar form of the complex number.