Answer
Verified
430.5k+ views
Hint: We can solve the above given question by applying the Euler’s formula. Euler’s formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.
Complete step by step answer:
Euler’s formula states that for any real number $x$,${{e}^{ix}}=\cos x+i\sin x$
Where $e$ is the base of the natural logarithm, $i$ is the imaginary unit, and $\cos $ and $\sin $ are the trigonometric functions sine and cosine respectively.
This complex exponential function is sometimes denoted by $cis\text{ x}$. The formula is still valid if $\text{ x}$ is a complex number.
Now according to the given question we have to prove that, ${{e}^{i\pi }}=-1$
Now from the Euler’s formula we can solve it.
As we have been already discussed earlier Euler’s formula is, ${{e}^{ix}}=\cos x+i\sin x$
$\Rightarrow {{e}^{i\pi }}=\cos \pi +i\sin \pi $
Now by using the trigonometric values of $\sin \pi =0\text{ and cos}\pi \text{=-1}$, we get,
$\Rightarrow {{e}^{i\pi }}=-1+0$
$\Rightarrow {{e}^{i\pi }}=-1$
Hence, proved
Now, as it have been asked in the question now we have to calculate ${{e}^{i\dfrac{\pi }{4}}}$
Now we can solve this also by using the Euler’s formula.
As we have been already discussed earlier Euler’s formula is ${{e}^{ix}}=\cos x+i\sin x$
Now by using the Euler’s formula we can solve this one.
${{e}^{i\dfrac{\pi }{4}}}=\cos \dfrac{\pi }{4}+i\sin \dfrac{\pi }{4}$
Now by using the trigonometric values of $\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}\text{ and sin}\dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}$ we can solve the complex equation. Now substitute the values of sine and cosine in the complex equation. By substituting the values we get,
${{e}^{i\dfrac{\pi }{4}}}=\dfrac{1}{\sqrt{2}}\text{ }+i\dfrac{1}{\sqrt{2}}\text{ }$
$\Rightarrow {{e}^{i\dfrac{\pi }{4}}}=\dfrac{1+i}{\sqrt{2}}\text{ }$
Therefore we can conclude that ${{e}^{i\dfrac{\pi }{4}}}=\dfrac{1+i}{\sqrt{2}}\text{ }$ .
Note: We should be careful while doing the complex numbers. We should be well aware of the complex numbers and Euler’s formula and its usage. The Euler’s formula states that the value of ${{e}^{i\theta }}$ is $\cos \theta +i\sin \theta $ where $\theta $ is the argument we can say that any complex number can be expressed as $z=a+ib=r{{e}^{i\theta }}$ where $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$ .
Complete step by step answer:
Euler’s formula states that for any real number $x$,${{e}^{ix}}=\cos x+i\sin x$
Where $e$ is the base of the natural logarithm, $i$ is the imaginary unit, and $\cos $ and $\sin $ are the trigonometric functions sine and cosine respectively.
This complex exponential function is sometimes denoted by $cis\text{ x}$. The formula is still valid if $\text{ x}$ is a complex number.
Now according to the given question we have to prove that, ${{e}^{i\pi }}=-1$
Now from the Euler’s formula we can solve it.
As we have been already discussed earlier Euler’s formula is, ${{e}^{ix}}=\cos x+i\sin x$
$\Rightarrow {{e}^{i\pi }}=\cos \pi +i\sin \pi $
Now by using the trigonometric values of $\sin \pi =0\text{ and cos}\pi \text{=-1}$, we get,
$\Rightarrow {{e}^{i\pi }}=-1+0$
$\Rightarrow {{e}^{i\pi }}=-1$
Hence, proved
Now, as it have been asked in the question now we have to calculate ${{e}^{i\dfrac{\pi }{4}}}$
Now we can solve this also by using the Euler’s formula.
As we have been already discussed earlier Euler’s formula is ${{e}^{ix}}=\cos x+i\sin x$
Now by using the Euler’s formula we can solve this one.
${{e}^{i\dfrac{\pi }{4}}}=\cos \dfrac{\pi }{4}+i\sin \dfrac{\pi }{4}$
Now by using the trigonometric values of $\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}\text{ and sin}\dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}$ we can solve the complex equation. Now substitute the values of sine and cosine in the complex equation. By substituting the values we get,
${{e}^{i\dfrac{\pi }{4}}}=\dfrac{1}{\sqrt{2}}\text{ }+i\dfrac{1}{\sqrt{2}}\text{ }$
$\Rightarrow {{e}^{i\dfrac{\pi }{4}}}=\dfrac{1+i}{\sqrt{2}}\text{ }$
Therefore we can conclude that ${{e}^{i\dfrac{\pi }{4}}}=\dfrac{1+i}{\sqrt{2}}\text{ }$ .
Note: We should be careful while doing the complex numbers. We should be well aware of the complex numbers and Euler’s formula and its usage. The Euler’s formula states that the value of ${{e}^{i\theta }}$ is $\cos \theta +i\sin \theta $ where $\theta $ is the argument we can say that any complex number can be expressed as $z=a+ib=r{{e}^{i\theta }}$ where $r=\sqrt{{{a}^{2}}+{{b}^{2}}}$ .
Recently Updated Pages
Who among the following was the religious guru of class 7 social science CBSE
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Trending doubts
A rainbow has circular shape because A The earth is class 11 physics CBSE
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE