Determine the polar form of the complex number 3 - 2i. Express the angle theta in degrees, where 0 < theta < 360°, and round numerical entries in the answer to two decimal places.
326.31(cos 326.31° – isin 326.31°)
3.61(cos 326.31° – isin 326.31°)
326.31(cos 3.61° + isin 3.61°)
3.61(cos 326.31° + isin 326.31°)
r = √13, so it must be b or d
assuming that the angles are correct, the number is in IV, so (d) is the choice.
(since sin 326.31° < 0)
To determine the polar form of a complex number, we need to express it in terms of its magnitude (also known as modulus or absolute value) and its argument (angle).
The magnitude of a complex number z = a + bi is given by the formula |z| = sqrt(a^2 + b^2).
In this case, the complex number is 3 - 2i. So, the magnitude is |3 - 2i| = sqrt(3^2 + (-2)^2) = sqrt(9 + 4) = sqrt(13) ≈ 3.61.
To find the argument of a complex number, we use the formula theta = atan2(b, a), where atan2 is the arc tangent function that takes into account the signs of the real and imaginary parts.
In this case, the real part is 3 and the imaginary part is -2. So, the argument is theta = atan2(-2, 3) ≈ -0.588.
To convert the argument to degrees, we multiply it by 180/π. Therefore, theta ≈ -0.588 * 180/π ≈ -33.69°.
Since we want the angle to be in the range (0, 360°), we add 360° to the angle if it is negative. In this case, theta ≈ -33.69° + 360° ≈ 326.31°.
Putting it all together, the polar form of the complex number 3 - 2i is approximately 3.61(cos 326.31° + isin 326.31°).