00:01
In subpart a of this problem we are provided with the complex number z equals to 2 minus i and we are asked to write it in the exponential form that is z equals to r times e raised to the power i teta where r is given by square root of x squared plus y squared and teta is given by making use of alpha which equals to tan inverse of the modulus of y over x.
00:30
So this is the value of alpha.
00:33
From here, the value of theta can be determined based on the signs of x and y.
00:43
So now let us begin by finding out r.
00:46
So r equals to square root of x squared, which is 2 squared, that is 4, plus negative 1 the whole squared, that is 1.
00:55
So we get r to be equal to square root of 5.
00:58
Next we calculate theta.
00:59
So, teta is given by alpha, alpha equals to tan inverse of the model is of y which is negative 1 over x which is 2.
01:11
So this equals to tan inverse of 1 over 2.
01:15
So here, since we have y to be negative and we have x to be positive, which implies that the complex number lies in the fourth quadrant and and in the fourth quadrant, we know that tan is negative.
01:34
So we have teta to be equal to negative tan inverse of 1 over 2.
01:39
So making use of this, we have the complex number z to be square root of 5 times e raised to the power negative i tan inverse of 1 over 2.
01:53
And therefore this is the required complex number in the exponential form.
01:59
So therefore this is the required answer for subpart a...