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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.

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Roots of integers

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.

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