If (1+2 i)/(2+i)=r[cosθ+i sinθ], then (a) r=1 ; θ=tan^-1 (3)/(4) (b) r=√(5) ; θ=tan^-1 (4)/(3) (

If (1+2 i)/(2+i)=r[cosθ+i sinθ], then (a) r=1 ; θ=tan^-1...

Key Concepts

Complex Numbers

Complex numbers extend the real numbers by including the imaginary unit i, where i² = -1. They allow for the representation of all numbers in the form a + bi, unifying the treatment of real and imaginary components in algebra and geometry.

Division of Complex Numbers

Dividing complex numbers involves removing the imaginary unit from the denominator, typically by multiplying the numerator and denominator by the complex conjugate of the denominator. This process yields a result in standard form that can then be manipulated or transformed into other representations.

Polar Form Representation

Complex numbers can be expressed in polar form as r(cos ? + i sin ?), where r, the modulus, measures the distance from the origin, and ?, the argument, represents the angle made with the positive real axis. This form is especially useful for multiplication and division since it simplifies operations to manipulations of r and ?.

Modulus and Argument Calculation

The modulus of a complex number, calculated as the square root of the sum of the squares of its real and imaginary parts, represents its magnitude. The argument, which is determined using inverse trigonometric functions (commonly the arctan of the ratio of the imaginary part to the real part), describes the angle of the complex number relative to the positive real axis.

Transcript

00:01 So in the given question we are told that if 1 plus 2i, if 1 plus 2i divided by 2 plus i is equal to r times cost theta plus i sine theta then cost theta plus i side theta then which of the following options is correct.

00:34 So option a is saying that r is equal to 1 and theta is equal to tan inverse of 3 by 4.

00:45 Option b is saying that r is equal to square root of 5 and theta is equal to tan inverse of 4 by 3.

00:58 Option c is saying r is equal to 1 and theta is equal to tan inverse of 4 by 3 option d is saying r is equal to 1 and theta is equal to tan inverse of minus 3 by 4 right so these are the options that are given in the question.

01:30 So let's start with the expression that is given over here.

01:35 So what we have is 1 plus 2 i divided by 2 plus i right? so let's multiply both the numerator and the denominator of this expression with 2 minus i so that we can simplify the expression as we can write this as i can write this as 1 times 2 is 2 plus 2 times 2i is 4i minus i minus 2 i squared divided by 2 square minus i square minus i square right and now this is equal to 2 plus 3 i minus i square is equal to 2 plus 3 i minus i square is equal to minus 1 so we will have minus 2 times i square which is equal to 2 and this is divided by 4 minus minus 1 so what we will have is 4 plus 3 i divided by 5 right and we can write this in the form 4 by 5 plus 3 by 5 i is equal to r times cost theta plus i sine theta and on expanding the right hand side we have r cost theta plus i times r sine theta right and over here we need to find the value of r and teta, right? so when we take tan theta from this, when we compare these expressions, we can see that, we can see that over here, sine theta r cost theta is equal to 4 by 5 and r r sine theta is equal to 3 by 5.

04:00 So when taking tan theta, well when we take r sine theta divided by r cost theta we will have tan theta which is equal to sine theta, our sine theta is 3 by 5 and r cost theta is 4 by 5.

04:21 So tan theta would be equal to 3 by 5 divided by 4 by 5 which is equal to 3 by 4.

04:31 And from this we can write that theta is equal to tan inverse of 3 by 4.

04:40 So we found the the part the theta part of the answer and it is equal to tan inverse of 3 by 4.

04:49 Right and as for r we can find r by taking we can find r by taking r coss theta square plus r sine theta square and this would be equal to 4 by 5 square plus 3 by 5 square and this is equal to r square cost square theta plus r square sine square theta that is equal to 16 by 5 25 plus 9 by 25 and from this we can write r square co square theta plus sine square theta is equal to 25 divided by 25 which is equal to 1 and we know that cos square theta plus s.

05:54 S...

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