Question

How do you evaluate \[{{e}^{\dfrac{\pi }{12}i}}-{{e}^{\dfrac{13\pi }{8}i}}\] using trigonometric functions?

Answer 1

Hint: For evaluating the expression given in the above question, we need to use the Euler’s identity, which is given by ${{e}^{i\theta }}=\cos \theta +i\sin \theta $. On substituting $\theta =\dfrac{\pi }{12}$ into this identity, $\theta =\dfrac{13\pi }{8}$ we can convert the given polar form of the complex expression into the rectangular complex form. Then using the different trigonometric identities and on substituting the values of the trigonometric functions for the different values of the angles, we will obtain the final simplified complex expression in the standard form of $a+ib$.

Complete step by step answer:
Let us consider the complex expression given in the above question as
\[\Rightarrow E={{e}^{\dfrac{\pi }{12}i}}-{{e}^{\dfrac{13\pi }{8}i}}.........\left( i \right)\]
From the Euler’s identity, we know that
$\Rightarrow {{e}^{i\theta }}=\cos \theta +i\sin \theta $
On substituting $\theta =\dfrac{\pi }{12}$ into the above identity, we get
$\Rightarrow {{e}^{\dfrac{\pi }{12}i}}=\cos \dfrac{\pi }{12}+i\sin \dfrac{\pi }{12}........\left( ii \right)$
Similarly, on substituting $\theta =\dfrac{13\pi }{8}$ we get
$\Rightarrow {{e}^{\dfrac{13\pi }{8}i}}=\cos \dfrac{13\pi }{8}+i\sin \dfrac{13\pi }{8}........\left( iii \right)$
Now, we substitute the equations (ii) and (iii) into the equation (i) to get
\[\begin{align}
  & \Rightarrow E=\cos \dfrac{\pi }{12}+i\sin \dfrac{\pi }{12}-\left( \cos \dfrac{13\pi }{8}+i\sin \dfrac{13\pi }{8} \right) \\
 & \Rightarrow E=\cos \dfrac{\pi }{12}+i\sin \dfrac{\pi }{12}-\cos \dfrac{13\pi }{8}-i\sin \dfrac{13\pi }{8} \\
\end{align}\]
On substituting $\cos \dfrac{\pi }{12}=0.96$ and $\sin \dfrac{\pi }{12}=0.26$ above, we get
\[\Rightarrow E=0.96+0.26i-\cos \dfrac{13\pi }{8}-i\sin \dfrac{13\pi }{8}\]
Writing $\dfrac{13\pi }{8}=\pi +\dfrac{5\pi }{8}$ in the above equation, we get
\[\Rightarrow E=0.96+0.26i-\cos \left( \pi +\dfrac{5\pi }{8} \right)-i\sin \left( \pi +\dfrac{5\pi }{8} \right)\]
Now, we know that \[\cos \left( \pi +\theta \right)=-\cos \theta \] and \[\sin \left( \pi +\theta \right)=-\sin \theta \]. Therefore, we can write the above expression as
\[\Rightarrow E=0.96+0.26i+\cos \dfrac{5\pi }{8}+i\sin \dfrac{5\pi }{8}\]
Finally, on substituting $\cos \left( \dfrac{5\pi }{8} \right)=-0.38$ and $\sin \left( \dfrac{5\pi }{8} \right)=0.92$ in the above expression, we get
\[\begin{align}
  & \Rightarrow E=0.96+0.26i-0.38+0.92i \\
 & \Rightarrow E=0.58+1.18i \\
\end{align}\]

Hence, we have finally simplified the given expression as \[0.58+1.18i\].

Note: For solving these types of questions we must remember the Euler’s identity which is given as ${{e}^{i\theta }}=\cos \theta +i\sin \theta $. Also, since we are dealing with the trigonometric functions, we must be comfortable with the different trigonometric identities. We must note that the values such as $\cos \left( \dfrac{5\pi }{8} \right)=-0.38$, $\sin \left( \dfrac{5\pi }{8} \right)=0.92$ etc . are found with the help of the scientific calculator and cannot be memorized.