Answer:
- The modulus of z is approximately 1.25.
- The argument of z is approximately 0.296.
Explanation:
To find the modulus and argument of the complex number z = 1 + cos(6π/5) + i sin(6π/5), we can use the trigonometric form of a complex number.
The modulus of a complex number z = a + bi is given by |z| = sqrt(a^2 + b^2).
In this case, the real part (a) is 1 and the imaginary part (b) is sin(6π/5). Therefore, the modulus of z is:
|z| = sqrt(1^2 + sin^2(6π/5))
To find the argument of a complex number, we use the angle between the positive real axis and the line connecting the origin and the complex number in the complex plane. The argument is given by arg(z) = tan^(-1)(b/a).
In this case, the argument of z is:
arg(z) = tan^(-1)(sin(6π/5)/1)
Now, let's calculate the modulus and argument:
|z| = sqrt(1 + sin^2(6π/5))
|z| ≈ sqrt(1 + 0.309^2) ≈ 1.25
arg(z) = tan^(-1)(sin(6π/5)/1)
arg(z) ≈ tan^(-1)(0.309/1) ≈ 0.296
To summarize:
- The modulus of z is approximately 1.25.
- The argument of z is approximately 0.296.