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Find the argument of the complex number two minus seven π in radians.
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Give your answer correct to two decimal places.
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We have been given a complex number in rectangular or algebraic form.
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In general, we can say a complex number is in this form if it is π plus ππ.
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And weβre being asked to find its argument.
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Weβll begin by considering what the number two minus seven π looks like on the Argand diagram.
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Remember, this is a way of representing complex numbers graphically.
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We have the horizontal axis, which represents the real component of our number.
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And the vertical axis represents its imaginary part.
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We can see then that the number two minus π§ [seven] π must lie in the fourth quadrant.
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And if we join this point to the origin, the argument is the angle this line makes with the horizontal.
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In fact, we measure this in a counterclockwise direction.
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So this tells us the value of our argument, letβs call that π, is going to be negative.
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And we know itβs negative rather than a large value of π because we generally represent our complex numbers using the principal argument.
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Thatβs π is greater than negative π and less than or equal to π.
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Letβs now add a right-angle triangle.
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We can see that the side adjacent to the included angle π must be two units.
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And the side opposite from the included angle is seven units.
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And since this is a right-angle triangle for which we know the measure of two of its sides and weβre looking to find the missing angle, we can use right-angle trigonometry.
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Here, tan π is equal to opposite over adjacent.
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So we can say that, for our value of π, tan π is equal to seven over two.
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And we solve this equation for π by finding the inverse tan of both sides of the equation.
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The inverse tan of tan π or the arctan of tan π is π.
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So we can see that π is equal to the inverse tan of seven over two.
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And as long as we have our calculator working in radians, we get π to be equal to 1.2924 and so on.
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And we said that our value of π needs to be negative.
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So π is negative 1.29 radians.
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Now, this is a rather long-winded method.
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And in fact, we can generalize it for the complex number of the form π plus ππ.
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We say that, for this complex number, its argument is the arctan or inverse tan of π divided by π.
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Letβs see what that looks like with our complex number.
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The constant π or the real part is two.
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And the coefficient of π or the imaginary component is negative seven.
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π is negative seven.
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So in this case, we say that π is equal to inverse tan of negative seven over two.
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And if we type that into our calculator, we get negative 1.29 radians.
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Itβs much more sensible to use this formula when finding the argument of the complex number.
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But we can use an Argand diagram to check our answer.
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In this case, π or the argument of our complex number is negative 1.29 radians.