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Find the power of the complex number: Leave the result in polar form[0.4(cos 850 sin 852)14[0.4(cos 85? +jsin 852)14 = [ (Simplify your answer Use integers or decim...

Question

Find the power of the complex number: Leave the result in polar form[0.4(cos 850 sin 852)14[0.4(cos 85? +jsin 852)14 = [ (Simplify your answer Use integers or decimals rounded to three decimal places as needed for any numbers in the expression. Type your answer in degrees Express complex numbers in terms of j Do not include the degree symbol in your answer:)

Find the power of the complex number: Leave the result in polar form [0.4(cos 850 sin 852)14 [0.4(cos 85? +jsin 852)14 = [ (Simplify your answer Use integers or decimals rounded to three decimal places as needed for any numbers in the expression. Type your answer in degrees Express complex numbers in terms of j Do not include the degree symbol in your answer:)



Answers

Find the product of the complex numbers. Leave answers in polar form.
$$z_{1}=4\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)$$
$$z_{2}=7\left(\cos 25^{\circ}+i \sin 25^{\circ}\right)$$

We have a complex number. We have to find it. Fourth power. Here we can use theme over there. It states that if it's it is a complex number and it's trigonometry form City called to our coast teacher plus I sine theta. Then it's and the power is given by that Raised to Ennis are raised to end in do cause and teacher plus I sine ended. So here we have a complex number in the techno metric form. We have to find it for power, said Rested for us three days to foreign do course 15 in before plus I sine 15 in do for that is equal to it even in do because 60 plus ice I in 60 that is 81 in do one by two. Plus I endure three by two, which is equal to 81 by two. Plus, I didn't do train do it 81 bye to the sister standard form of parasols

Were given these two complex numbers and we want to find the product of those We wanna find Z one times Z too. Come on, you're in public Polar farm. We just acted to operations. We have to take the radius from each of our complex numbers and multiply them together. So three times 10 gives May 30 and then for the drink functions here we have to take the angles and we have to add those together. So we have to do five pie over eight plus pi over 16 which means we need a common denominator, which is 16. So will multiply this first fraction by two over too to make that 10 pie over 16 plus pi over 16. So that will give us go sign of 11 pie over 16 plus I times a sign of the 11 pie over 16

Okay, We're giving these to complex numbers and they want us to find the product of those too complex numbers. Now, the product is simply taking the two rady I that we have here and we're going to multiply them together and then taking the angle measurements that we have and we add those together. So Z one times Z too is gonna be six times five, which is 30 times a quantity. Co sign off had my two angles 20 degrees plus 50 degrees is 70 degrees plus i times a sign of again having the angles 70 degrees.

Okay, were given this complex number in polar form and were asked to raise it to the third power. And we're gonna use D move race theorem to do that. And it's the nice part about polar form. It is a whole lot easier to raise complex numbers to powers in polar form than it is in rectangular form. Because what we have to do is take the power and apply it to our module iss to the radius that we have. So we're gonna have four to the third power, and then we have to take the value of our power and we multiply the angle arguments by that we don't raise the angles to the power. We multiply that by the power. So we're gonna have Ford is the third power, which is 64 times Call sign off three times 15 which it iss 45 degrees plus i times a sign, huh? 45 degrees. And so there we have our answer for this race to the third power. Now, I also want us to then write this in rectangular form. So that means we have to turn our call sign of 45 sign of 45 into their values and distribute the 64 through there. So we're gonna have co sign of 45 is the square root of two over, too. And the sign of 45 is also the square to over two. And when we distribute that 64 times, the square to 2/2 is going to give us 32 times the square root of two, plus 32 high times the square root of two.


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