Express this in the form?

Question

Express 1+i in the form R(cos θ+i sin θ)=Re^iθ.

Answer: √2e^{i[(pi/4)+2m*pi]}

 

Answer 1

1+i

= √2[1/√2 + i/√2]

= √2[cos(pi/4 + 2m*pi) + i sin(pi/4 + 2m*pi)], where m can be any integer.

= √2e^{i[(pi/4)+2m*pi]}

-------

Ideas: A+iB = √(A^2 + B^2) [A/√(A^2 + B^2) + iB/√(A^2 + B^2)], and cos(pi/4 + 2m*pi) = sin(pi/4 + 2m*pi) = 1/√2

Answer 2

Think of 1 + i as a vector on the complex plane; when the problem is asking you to express it in the form Re^iθ, it is asking for the magnitude R and angle θ. R can be easily calculated through the Pythagorean Theorem as √(1^2 + 1^2) or √2, while the tangent of the angle θ is equal to the imaginary value divided by the real value (here, we treat i as 1, 2i as 2, etc.), so it is 1/1 or 1. θ, then, is tan^-1(1).

For the great majority of these problems, you're asked to provide the angle in radians, so the angle should be pi/2. Therefore, your answer should be √2e^(i*pi/2).

Answer 3

For the complex expression    1+i

Calculate R = √1²+1² = √2 , tan θ = 1/1 = 1, and θ=π/4 = Argument Z, where π = pi

Re^iθ = √2e^iπ/4, Now recognizing that the Argument can have multiple values for every complete revolution of 2π in a trigonometric circle, the argument can be expressed in terms of multiples of 2π such

that θ = π/4 +2kπ where k = 1,2,3,4.....n

Therefore the general form of the solution is √2e^i(π/4+2kπ)