From euler’s relation eiθ = cosθ + isinθ,

(a) find the geometric representation of e-iθ, (b) express cosθ in terms of exponentials, (c) express isinθ in terms of exponentials.

(a) The graphic representation is in the attached figure.

(b) .

(c) .

Step-by-step explanation:

(a) Given a complex number we know, from Euler's formula that . So, it is not difficult to notice that

so it is on the unit circumference. Also, notice that the Cartesian representation of the complex number is .

Now,

.

Notice that has the same modulus that , so it is on the unit circumference. Beside, its Cartesian representation is .

So, the points and are symmetric with respect to the X-axis. All this can be checked in the attached figure.

(b) Notice that

Then,

.

(c) Notice that

Then,

.

answer: the answer is 23

step-by-step explanation:

circumference divided by 3.141592654 then timesthe radius twice

step-by-step explanation:

circumference divided by 3.141592654 then timesthe radius twice