Question

How do you evaluate ${e^{\dfrac{\pi }{4}i}} - {e^{\dfrac{{3\pi }}{4}i}}$ using trigonometric function?

Answer 1

According to the given in the question we have to determine the value or evaluate ${e^{\dfrac{\pi }{4}i}} - {e^{\dfrac{{3\pi }}{4}i}}$ using trigonometric function. So, to evaluate the given trigonometric function first of all we have to determine the value of the trigonometric term which ${e^{\dfrac{\pi }{4}i}}$with the help of the formula as mentioned below:

Formula used:
$\Rightarrow {e^{\dfrac{\pi }{4}i}} = \cos \dfrac{\pi }{4} + i\sin \dfrac{\pi }{4}.................(A)$
Now, same as to evaluate the given trigonometric function we have to determine the value of the term of the given trigonometric function which is ${e^{\dfrac{{3\pi }}{4}i}}$with the help of the formula as mentioned below:

$\Rightarrow {e^{\dfrac{{3\pi }}{4}i}} = \cos \dfrac{{3\pi }}{4} + i\sin \dfrac{{3\pi }}{4}.................(B)$
Now, as we have already obtained the values of the both of the terms with the help of the formulas (A) and (B) we have to substitute those values in that given trigonometric function to evaluate.
Now, to solve the obtained expression we have to use the formulas as mentioned below:

$\Rightarrow \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}...........(C) \\\ \Rightarrow \operatorname{Sin} \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}...........(D) \\\$

Complete step by step answer:
Step 1: First of all we have to determine the value of the trigonometric term which ${e^{\dfrac{\pi }{4}i}}$with the help of the formula (A) as mentioned in the solution hint. Hence,
$\Rightarrow {e^{\dfrac{\pi }{4}i}} = \cos \dfrac{\pi }{4} + i\sin \dfrac{\pi }{4}..................(1)$
Step 2: Now, same as the solution step 1 we have to evaluate the given trigonometric function we have to determine the value of the term of the given trigonometric function which is ${e^{\dfrac{{3\pi }}{4}i}}$with the help of the formula (B) as mentioned in the solution hint. Hence,
$\Rightarrow {e^{\dfrac{{3\pi }}{4}i}} = \cos \dfrac{{3\pi }}{4} + i\sin \dfrac{{3\pi }}{4}..................(2)$
Step 3: Now, as we have already obtained the values of the both of the terms with the help of the formulas (A) and (B) we have to substitute those values in that given trigonometric function${e^{\dfrac{\pi }{4}i}} - {e^{\dfrac{{3\pi }}{4}i}}$ to evaluate it. Hence, on substituting both of the expression (1) and (2) in${e^{\dfrac{\pi }{4}i}} - {e^{\dfrac{{3\pi }}{4}i}}$.

$\Rightarrow \cos \dfrac{\pi }{4} + i\sin \dfrac{\pi }{4} - \cos \dfrac{{3\pi }}{4} + i\sin \dfrac{{3\pi }}{4}$………………….(3)
Step 4: Now, we have to solve the expression (3) with the help of the formulas (C) and (D) which are as mentioned in the solution hint. Hence,
$\Rightarrow \dfrac{{1 + i}}{{\sqrt 2 }} - \dfrac{{ - 1 + i}}{{\sqrt 2 }}$
On solving the expression as obtained just above,
$= \dfrac{{1 + i + 1 - i}}{{\sqrt 2 }} \\\ = \dfrac{2}{{\sqrt 2 }} \\\$

Hence, with the help of the formula (A), (B), (C), and (D) we have evaluated the given trigonometric expression${e^{\dfrac{\pi }{4}i}} - {e^{\dfrac{{3\pi }}{4}i}} = \dfrac{2}{{\sqrt 2 }}$.

Note: To evaluate the whole expression it is necessary that we have to evaluate the values of ${e^{\dfrac{\pi }{4}i}}$ and ${e^{\dfrac{{3\pi }}{4}i}}$ with the help of the formulas (A) and (B) which are mentioned in the solution hint.
We can determine the value of $\cos \dfrac{{3\pi }}{4}$ as be finding the value of $- \cos \dfrac{\pi }{4}$with the help of the formulas (C) and (D) as mentioned in the solution hint.