Answer
Verified
428.1k+ views
Hint:- \[\operatorname{Re} (z)\]is the real part of \[z\] and \[\operatorname{Im} (z)\] is the imaginary part of \[z\].
If \[z\] is any complex number then it can be written as,
\[ \Rightarrow z = x + iy = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )\] (1)
As we know that \[\operatorname{Re} (z) = x,\operatorname{Im} (z) = y{\text{ and }}|z| = r\]
Using equation 1 we can write,
\[ \Rightarrow |x| + |y| = r[|\cos \theta | + |\sin \theta |]\] (2)
Now, to prove the given relation.
On squaring equation 2 we get,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[{\cos ^2}\theta + {\sin ^2}\theta + |2\sin \theta \cos \theta |]\]
As we know that, \[{\cos ^2}\theta + {\sin ^2}\theta = 1\].
So, solving above equation it becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |2\sin \theta \cos \theta |]\]
As we know that according to trigonometric identities,
\[2\sin \theta \cos \theta = \sin 2\theta \]
So, on solving the above equation. It becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |\sin 2\theta |]\]
Now, as we know that, (3)
\[ \Rightarrow {\text{|}}\sin x| \leqslant 1\]
So, the maximum value of \[{\text{sin2}}\theta = 1\].
Now , equation 3 becomes,
\[ \Rightarrow {[|x| + |y|]^2} \leqslant 2{r^2}\]
Now, taking square roots to both sides of the above equation. We get,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 r\]
So, above equation can be written as,
\[ \Rightarrow |\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant \sqrt 2 |z|\]
Using equation 1 above equation can be written as,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 |x + iy|\]
Hence, \[|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 {\text{ }}\]or \[|x| + |y| \leqslant \sqrt 2 |x + iy|\]
Note:- Whenever you came up with this type of problem then easiest and efficient way
is to write complex number \[z\], in polar form \[\left( {z = r{e^{i\theta }}} \right)\], in terms of \[{\text{sin}}\theta \]and \[\cos \theta \]\[\left( {z = r(\cos \theta + i\sin \theta )} \right)\], or in terms of x and y \[\left( {z = x + iy} \right)\] as per required result to be proved.
If \[z\] is any complex number then it can be written as,
\[ \Rightarrow z = x + iy = r{e^{i\theta }} = r(\cos \theta + i\sin \theta )\] (1)
As we know that \[\operatorname{Re} (z) = x,\operatorname{Im} (z) = y{\text{ and }}|z| = r\]
Using equation 1 we can write,
\[ \Rightarrow |x| + |y| = r[|\cos \theta | + |\sin \theta |]\] (2)
Now, to prove the given relation.
On squaring equation 2 we get,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[{\cos ^2}\theta + {\sin ^2}\theta + |2\sin \theta \cos \theta |]\]
As we know that, \[{\cos ^2}\theta + {\sin ^2}\theta = 1\].
So, solving above equation it becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |2\sin \theta \cos \theta |]\]
As we know that according to trigonometric identities,
\[2\sin \theta \cos \theta = \sin 2\theta \]
So, on solving the above equation. It becomes,
\[ \Rightarrow {[|x| + |y|]^2} = {r^2}[1 + |\sin 2\theta |]\]
Now, as we know that, (3)
\[ \Rightarrow {\text{|}}\sin x| \leqslant 1\]
So, the maximum value of \[{\text{sin2}}\theta = 1\].
Now , equation 3 becomes,
\[ \Rightarrow {[|x| + |y|]^2} \leqslant 2{r^2}\]
Now, taking square roots to both sides of the above equation. We get,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 r\]
So, above equation can be written as,
\[ \Rightarrow |\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant \sqrt 2 |z|\]
Using equation 1 above equation can be written as,
\[ \Rightarrow |x| + |y| \leqslant \sqrt 2 |x + iy|\]
Hence, \[|\operatorname{Re} (z)| + |\operatorname{Im} (z)| \leqslant |z|\sqrt 2 {\text{ }}\]or \[|x| + |y| \leqslant \sqrt 2 |x + iy|\]
Note:- Whenever you came up with this type of problem then easiest and efficient way
is to write complex number \[z\], in polar form \[\left( {z = r{e^{i\theta }}} \right)\], in terms of \[{\text{sin}}\theta \]and \[\cos \theta \]\[\left( {z = r(\cos \theta + i\sin \theta )} \right)\], or in terms of x and y \[\left( {z = x + iy} \right)\] as per required result to be proved.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE