Answer:
a. To find the trigonometric form of the complex number -5 + 3i, we can use the formula r(cosθ + isinθ), where r is the magnitude (distance from the origin) and θ is the argument (angle with the positive real axis).
The magnitude can be found using the Pythagorean theorem:
|z| = √((-5)^2 + 3^2)
|z| = √(25 + 9)
|z| = √34
To find the argument θ, we can use the inverse tangent function:
θ = tan^(-1)(3 / -5)
θ ≈ -0.588 radians
Therefore, the trigonometric form of the complex number -5 + 3i is approximately √34(cos(-0.588) + isin(-0.588)).
b. The complex number 5 can be written as 5(cos0 + isin0) since the magnitude is 5 and the argument is 0 radians.
c. The complex number 5(cosπ + isinπ) can also be written as -5 since the magnitude is 5 and the argument is π radians.
d. The complex number (cos(2/3π) + sin(2/3π)) can be written in trigonometric form as 1(cos(2/3π) + isin(2/3π)).
e. The complex number cos(4/π) + isin(4/π) can be written in trigonometric form as 1(cos(4/π) + isin(4/π)).
Note that in trigonometric form, the magnitude is positive and the argument is within the range of -π to π