Find the argument of the complex number -8-7i in the interval 0^(@) <= theta < 360^(@), rounding to the nearest tenth of a degree if necessary.

Identify parts: To find the argument of a complex number in the form z = a + bi, where a is the real part and b is the imaginary part, we use the formula theta = arctan(b/a). However, since the complex number is in the second or third quadrant (because the real part is negative), we need to add 180° to the result of arctan(b/a) to get the correct argument in the range 0° <= theta < 360°.Calculate arctan: First, identify the real part (a) and the imaginary part (b) of the complex number -8-7i. Here, a = -8 and b = -7.Find quadrant: Next, calculate the arctan(b/a) using the values of a and b. We have arctan(-7/-8). Since both a and b are negative, the complex number is in the third quadrant.Calculate correct argument: Using a calculator, find the value of arctan(-7/-8). The result is approximately arctan(0.875), which is about 41.2°. However, since the complex number is in the third quadrant, we need to add 180° to this result to find the correct argument.Add 180°: Add 180° to 41.2° to get the argument of the complex number in the correct range. So, the argument theta is 41.2° + 180° = 221.2°.

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Find the argument of the complex number
-8-7i in the interval
0^(@) <= theta < 360^(@), rounding to the nearest tenth of a degree if necessary.

Find the argument of the complex number $-8-7 i$ in the interval 0^{\circ} \leq \theta<360^{\circ} , rounding to the nearest tenth of a degree if necessary.

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Math Problems

Algebra 1

Domain and range of quadratic functions: equations

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Q. Find the argument of the complex number

-8-7 i

in the interval

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

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

Identify parts: To find the argument of a complex number in the form $z = a + bi$ , where $a$ is the real part and $b$ is the imaginary part, we use the formula $\theta = \text{arctan}(\frac{b}{a})$ . However, since the complex number is in the second or third quadrant (because the real part is negative), we need to add $180°$ to the result of $\text{arctan}(\frac{b}{a})$ to get the correct argument in the range 0° \leq \theta < 360°.
Calculate arctan: First, identify the real part $a$ and the imaginary part $b$ of the complex number $-8-7i$ . Here, $a = -8$ and $b = -7$ .
Find quadrant: Next, calculate the $\text{arctan}(\frac{b}{a})$ using the values of $a$ and $b$ . We have $\text{arctan}(\frac{-7}{-8})$ . Since both $a$ and $b$ are negative, the complex number is in the third quadrant.
Calculate correct argument: Using a calculator, find the value of $\arctan\left(-\frac{7}{-8}\right)$ . The result is approximately $\arctan(0.875)$ , which is about $41.2^\circ$ . However, since the complex number is in the third quadrant, we need to add $180^\circ$ to this result to find the correct argument.
Add $180$ °: Add $180°$ to $41.2°$ to get the argument of the complex number in the correct range. So, the argument $\theta$ is $41.2° + 180° = 221.2°$ .