[Here $i$ is iota]
Given the complex number $a=\cos\theta+i\sin\theta$. We need to find the value of $1+a^2$.
Finding $a^2$
Using the property of complex numbers in polar form (De Moivre's theorem), if $a = \cos\theta + i\sin\theta$, then $a^n = \cos(n\theta) + i\sin(n\theta)$.
Therefore, for $n=2$:
$a^2 = \cos(2\theta) + i\sin(2\theta)$
Calculating $1+a^2$
Now, substitute the expression for $a^2$ into $1+a^2$:
$1+a^2 = 1 + (\cos(2\theta) + i\sin(2\theta))$
$1+a^2 = (1 + \cos(2\theta)) + i\sin(2\theta)$
Applying Trigonometric Identities
Use the double angle identities:
- $1 + \cos(2\theta) = 2\cos^2\theta$
- $\sin(2\theta) = 2\sin\theta\cos\theta$
Substitute these identities back into the expression for $1+a^2$:
$1+a^2 = (2\cos^2\theta) + i(2\sin\theta\cos\theta)$
Factoring the Expression
Factor out the common term $2\cos\theta$ from both the real and imaginary parts:
$1+a^2 = 2\cos\theta(\cos\theta + i\sin\theta)$
This matches the expression given in Option A.
Important Questions from Miscellaneous
- Which of the following relationships is/are not true?
(A). Most probable velocity = $\sqrt{\frac{2RT}{M}}$
(B). PV = $\frac{3}{2}kT$
(C). Compressibility factor Z = $\frac{pV}{nRT}$
(D). Average kinetic energy of gas = $\frac{1}{2}kT$
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- Match List-I with List-II
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(A). Primary structure of protein is associated with amino acids
(B). Secondary structure of protein is associated to peptides
(C). Tertiary structure of protein is associated with polypeptide chains
(D). Quaternary structure of protein is associated with polypeptide chains
Choose the correct answer from the options given below:
