Find the argument of the complex number 8, minus, 5, i8−5i 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 8, minus, 5, i8−5i 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 argument of a complex number is the angle it makes with the positive real axis.
The complex number given is 8 - 5i.
We can find the argument (θ) using the formula θ = atan2(b, a), where a is the real part of the complex number and b is the imaginary part.
Here, a = 8 and b = -5.
So, θ = atan2(-5, 8).
This will give us the angle in radians. To convert it to degrees, we multiply by 180/π.
However, the atan2 function gives the result in the range -π to π, or -180° to 180°. If the result is negative, we add 360° to bring it into the range 0° to 360°.
Let's calculate:
θ = atan2(-5, 8) = -32.0054 degrees (after converting from radians to degrees)
Since this is negative, we add 360°:
θ = -32.0054 + 360 = 327.9946 degrees
Rounding to the nearest tenth of a degree, we get θ = 328.0°.
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