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Amiraneli [1.4K]
3 years ago
6

Express the complex number in trigonometric form. 5 - 5i

Mathematics
2 answers:
bazaltina [42]3 years ago
6 0

●✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎❀✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎●

               Hi my lil bunny!

❧⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯☙

Lets do this step by step.

This is the trigonometric form of a complex number where |z| is the modulus and 0 is the angle created on the complex plane.

z = a + bi = |z| (cos ( 0 ) + I sin (0))

The modulus of a complex number is the distance from the origin on the complex plane.

|z| = \sqrt{a^2 + b^2} where z = a + bi

Substitute the actual values of a = -5 and b = -5.

|z| = \sqrt{(-5) ^2 + (-5) ^2}

Now Find |z| .

Raise - 5 to the power of 2.

|z| = \sqrt{25 + (-5) ^2}

Raise - 5 to the power of 2.

|z| = \sqrt{25 + 25}

Add 25 and 25.

|z| = \sqrt{50}

Rewrite 50 as 5^2 . 2 .

|z| = 5\sqrt{2}

Pull terms out from under the radical.

|z| = 5\sqrt{2}

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

0 = arctan (\frac{-5}{-5} )

Since inverse tangent of  \frac{-5}{-5}  produces an angle in the third quadrant, the value of the angle is \frac{5\pi }{4} .

0 = \frac{5\pi }{4}

Substitute the values of 0 = \frac{5\pi }{4} and |z| = 5\sqrt{2} .

5\sqrt{2} ( cos( \frac{5\pi}{4})  + i sin (\frac{5\pi}{4}))

❧⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯☙

●✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎❀✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎✴︎●

Hope this helped you.

Could you maybe give brainliest..?

❀*May*❀

kolbaska11 [484]3 years ago
6 0

Answer:

five square root two times the quantity cosine of five pi divided by four plus i times sine of five pi divided by four

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