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Given that π is equal to negative 30 plus 30π, determine the principal amplitude of π to the fifth power.
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In this question, weβre given a complex number π written in algebraic form.
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Thatβs the form π plus ππ, where π and π are real numbers.
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We need to use this to determine the principal amplitude of π to the fifth power.
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To answer this question, letβs start by recalling what we mean by the principal amplitude of a complex number.
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The principal amplitude or principal argument of a complex number is the angle the line segment between π and the origin on an Argand diagram makes with the positive real axis, where we restrict this angle to be between negative π and π in radians and negative 180 and 180 degrees in degrees.
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In this question, weβre going to work in degrees.
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Therefore, to answer this question, weβre first going to need to determine the argument of the complex number π to the fifth power.
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Since we need to determine a complex number to an integer exponent, weβll do this by recalling de Moivreβs theorem.
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This tells us for a complex number written in trigonometric form, thatβs π times cos of π plus π sin of π, where π is greater than or equal to zero and π is any real number, then for any integer value of π, π times cos of π plus π sin of π all raised to the πth power is equal to π to the πth power multiplied by the cos of ππ plus π sin of ππ.
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In other words, when we raise a complex number to an integer exponent of π, we raise its magnitude to that value of π and we multiply its argument by π.
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To apply this to find π to the fifth power, weβre going to need to write π in trigonometric form.
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And to do this, we need to find the values of π and π which are the magnitude of π and the argument of π.
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Letβs start with the magnitude of π.
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Thatβs the distance between π and the origin on an Argand diagram.
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We can find this by finding the square root of the sums of the squares of the real and imaginary parts of π.
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Thatβs the square root of negative 30 squared plus 30 squared, which, if we evaluate, is equal to 30 root two.
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However, it is worth pointing out we donβt actually need to find this value.
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The magnitude of π tells us the distance between π and the origin in an Argand diagram.
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And we can see the magnitude of the complex number π does not affect its argument when raised to an integer exponent.
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Therefore, the magnitude of π will not affect the argument of π to the fifth power.
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And in particular, this means it wonβt affect its principal amplitude either.
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However, it can be useful to see how to write π in trigonometric form anyway.
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Next, we need to determine the argument of π.
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And weβll do this by first working out which quadrant in an Argand diagram π lies in.
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Since the real part of π is negative 30 and its imaginary part is 30, its π₯-coordinate will be negative 30 and its π¦-coordinate will be 30.
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This means it lies in the second quadrant.
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We can then determine the argument of π by recalling the following result.
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If π plus ππ is a complex number written in algebraic form in the second quadrant of an Argand diagram, then the argument of π plus ππ is equal to the inverse tan of π divided by π plus 180 degrees.
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This allows us to determine the argument of π.
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Our value of π, the imaginary part of π, is 30.
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And our value of π, the real part of π, is negative 30.
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So, the argument of π is the inverse tan of 30 divided by negative 30 plus 180 degrees.
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We can then evaluate this.
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The inverse tan of negative one is negative 45 degrees, giving us the argument of π is 135 degrees.
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We can then use this to write π in trigonometric form.
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π is 30 root two multiplied by the cos of 135 degrees plus π sin of 135 degrees.
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Now, we can use de Moivreβs theorem to raise both sides of the equation to the fifth power.
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Since five is an integer exponent, when we raise π to the fifth power, we raise its magnitude to the fifth power and we multiply its argument by five.
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π to the fifth power is 30 root two raised to the fifth power multiplied by the cos of five times 135 degrees plus π sin of five times 135 degrees.
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Now, weβre only interested in finding the principal amplitude of π to the fifth power.
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We can do this directly from its argument.
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First, weβll simplify this argument.
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Five multiplied by 135 degrees is 675 degrees.
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And now the principal argument or principal amplitude of this value is the equivalent angle between negative 180 and 180 degrees, including 180 degrees.
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We can then determine the principal amplitude of π to the fifth power by recalling both cosine and sine are periodic with a period of 360 degrees.
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In other words, we can add and subtract integer multiples of 360 degrees from its argument.
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Therefore, if we subtract 360 degrees from 675 degrees, we donβt change the value.
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This then gives us 315 degrees.
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However, this is not in the given interval.
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So we subtract another 360 degrees to get negative 45 degrees, which is in this interval.
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Therefore, we were able to show if π is negative 30 plus 30π, then the principal amplitude of π to the fifth power is negative 45 degrees.