Answer
Verified
400.5k+ views
Hint: Simplify the given expression using the formula ${{e}^{i\theta }}=\cos \theta +i\sin \theta $. Further use the fact that the product of an infinite sequence of increasing terms can take value 1 if and only if the value of each term is 1. Write equations based on this data and simplify the equation to calculate the value of $\theta $ which satisfies the given equation.
Complete step-by-step solution-
We have to prove that the general value of $\theta $ which satisfy the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ is $2\pi n$, where ‘n’ is an integer.
We know that ${{e}^{i\theta }}=\cos \theta +i\sin \theta $.
Thus, we can rewrite the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ as $\left( {{e}^{i\theta }} \right){{\left( {{e}^{i\theta }} \right)}^{2}}{{\left( {{e}^{i\theta }} \right)}^{3}}...=1$.
We know that $y={{e}^{i\theta }}$ is an increasing function whose range is always greater than or equal to zero.
We also know that the product of an infinite sequence of increasing terms can take value 1 if and only if the value of each term is 1.
Thus, we have ${{e}^{i\theta }}=1,{{\left( {{e}^{i\theta }} \right)}^{2}}=1,{{\left( {{e}^{i\theta }} \right)}^{3}}=1,...$.
So, we must have ${{e}^{i\theta }}=1$.
Thus, we have $\cos \theta +i\sin \theta =1$, which means $\cos \theta =1,\sin \theta =0$.
We will now calculate solutions to the equations $\cos \theta =1$ and $\sin \theta =0$.
We know that the general solutions of the equation $\cos \theta =1$ are $\theta =2n\pi ,n\in I$.
We also know that the general solutions of the equation $\sin \theta =0$ are $\theta =n\pi ,n\in I$.
So, the common solutions of the equations $\cos \theta =1$ and $\sin \theta =0$ is $\theta =2n\pi ,n\in I$.
Hence, we have proved that the general value of $\theta $ which satisfy the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ is $2\pi n$, where ‘n’ is an integer.
Note: We must know how to write general solutions of trigonometric equations; otherwise, we won’t be able to prove the given statement. We must also keep in mind that we must consider the solutions common to both the trigonometric equations.
Complete step-by-step solution-
We have to prove that the general value of $\theta $ which satisfy the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ is $2\pi n$, where ‘n’ is an integer.
We know that ${{e}^{i\theta }}=\cos \theta +i\sin \theta $.
Thus, we can rewrite the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ as $\left( {{e}^{i\theta }} \right){{\left( {{e}^{i\theta }} \right)}^{2}}{{\left( {{e}^{i\theta }} \right)}^{3}}...=1$.
We know that $y={{e}^{i\theta }}$ is an increasing function whose range is always greater than or equal to zero.
We also know that the product of an infinite sequence of increasing terms can take value 1 if and only if the value of each term is 1.
Thus, we have ${{e}^{i\theta }}=1,{{\left( {{e}^{i\theta }} \right)}^{2}}=1,{{\left( {{e}^{i\theta }} \right)}^{3}}=1,...$.
So, we must have ${{e}^{i\theta }}=1$.
Thus, we have $\cos \theta +i\sin \theta =1$, which means $\cos \theta =1,\sin \theta =0$.
We will now calculate solutions to the equations $\cos \theta =1$ and $\sin \theta =0$.
We know that the general solutions of the equation $\cos \theta =1$ are $\theta =2n\pi ,n\in I$.
We also know that the general solutions of the equation $\sin \theta =0$ are $\theta =n\pi ,n\in I$.
So, the common solutions of the equations $\cos \theta =1$ and $\sin \theta =0$ is $\theta =2n\pi ,n\in I$.
Hence, we have proved that the general value of $\theta $ which satisfy the equation $\left( \cos \theta +i\sin \theta \right){{\left( \cos \theta +i\sin \theta \right)}^{2}}{{\left( \cos \theta +i\sin \theta \right)}^{3}}...=1$ is $2\pi n$, where ‘n’ is an integer.
Note: We must know how to write general solutions of trigonometric equations; otherwise, we won’t be able to prove the given statement. We must also keep in mind that we must consider the solutions common to both the trigonometric equations.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE