Find the argument of the complex number minus, 7, plus, 7, i−7+7i in the interval 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding to the nearest tenth of a degree if necessary.
Question
Find the argument of the complex number minus, 7, plus, 7, i−7+7i in the interval 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ , rounding to the nearest tenth of a degree if necessary.
Solution
The complex number in question is -7 + 7i.
Step 1: Convert the complex number into polar form. The polar form of a complex number is r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the argument of the complex number.
Step 2: Calculate the magnitude r. r = sqrt((-7)^2 + 7^2) = sqrt(98) = 7 sqrt(2).
Step 3: Calculate the argument θ. θ = arctan(Imaginary part/Real part) = arctan(7/-7) = arctan(-1) = -45 degrees.
Step 4: Adjust the argument to fall within the specified interval [0, 360). Since our calculated argument is negative, we add 360 degrees to it to get the final argument. So, θ = -45 + 360 = 315 degrees.
So, the argument of the complex number -7 + 7i in the interval 0 ≤θ<360 is 315 degrees.
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