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Find the argument of the complex number in the interval .Express your answer in terms of .Answer:
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Q. Find the argument of the complex number in the interval .Express your answer in terms of .Answer:
- Determine angle for complex number: To find the argument of the complex number , we need to determine the angle that the line connecting the origin to the point makes with the positive -axis in the complex plane. The argument is the angle in the standard position (counter-clockwise from the positive -axis) that the radius vector makes with the positive -axis.
- Identify quadrant: The complex number is in the fourth quadrant of the complex plane because the real part is positive and the imaginary part is negative. In the fourth quadrant, the argument is related to the angle by , where is the angle the line makes with the negative -axis.
- Calculate angle : To find , we use the arctan function, which gives us the angle whose tangent is the ratio of the imaginary part to the real part of the complex number. However, since we are in the fourth quadrant, we need to take the positive value of the imaginary part to find the angle with respect to the negative x-axis. So, .
- Find : The is because the tangent of is . This is a well-known angle from trigonometry.
- Calculate final argument: Now we can find the argument by subtracting from : .
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