Find the argument of the complex number 6-6i in the interval 0^(@) <= theta < 360^(@), rounding to the nearest tenth of a degree if necessary.

Calculate arctan ratio: To find the argument of a complex number in the form a + bi, where a is the real part and b is the imaginary part, we use the formula theta = arctan(b/a). For the complex number 6-6i, a = 6 and b = -6.Adjust for quadrant: We calculate the arctan(-6/6) = arctan(-1). The arctan of -1 is -45°. However, the argument of a complex number should be expressed in the interval from 0° to 360°.Find final argument: Since the complex number is in the third quadrant (both real and imaginary parts are negative), we add 180° to the initial result to find the argument in the correct quadrant. Therefore, the argument theta = -45° + 180° = 135°.Check within interval: We check if the argument is within the specified interval 0° <= theta < 360°. Since 135° is within this interval, we do not need to adjust it further.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the argument of the complex number
6-6i in the interval
0^(@) <= theta < 360^(@), rounding to the nearest tenth of a degree if necessary.

Find the argument of the complex number $6-6 i$ in the interval 0^{\circ} \leq \theta<360^{\circ} , rounding to the nearest tenth of a degree if necessary.

View step-by-step help

Full solution

Q. Find the argument of the complex number

6-6 i

in the interval

0^{\circ} \leq \theta<360^{\circ}

, rounding to the nearest tenth of a degree if necessary.

Calculate arctan ratio: To find the argument of a complex number in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part, we use the formula $\theta = \arctan(\frac{b}{a})$ . For the complex number $6-6i$ , $a = 6$ and $b = -6$ .
Adjust for quadrant: We calculate the $\arctan(-\frac{6}{6}) = \arctan(-1)$ . The $\arctan$ of $-1$ is $-45°$ . However, the argument of a complex number should be expressed in the interval from $0°$ to $360°$ .
Find final argument: Since the complex number is in the third quadrant (both real and imaginary parts are negative), we add $180°$ to the initial result to find the argument in the correct quadrant. Therefore, the argument $\theta = -45° + 180° = 135°$ .
Check within interval: We check if the argument is within the specified interval 0^\circ \leq \theta < 360^\circ. Since $135^\circ$ is within this interval, we do not need to adjust it further.