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faltersainse [42]
4 years ago
8

Express the complex number in trigonometric form. -6 + 6\sqrt(3) i

Mathematics
1 answer:
horrorfan [7]4 years ago
4 0

Answer:

<h2>12(cos120°+isin120°)</h2>

Step-by-step explanation:

The rectangular form of a complex number is expressed as z = x+iy

where the modulus |r| = \sqrt{x^{2}+y^{2} and the argument \theta = tan^{-1}\frac{y}{x}

In polar form, x = rcos\theta \ and\ y = rsin\theta

z = rcos\theta+i(rsin\theta)\\z = r(cos\theta+isin\theta)

Given the complex number, z = -6+6\sqrt{3} i. To express in trigonometric form, we need to get the modulus and argument of the complex number.

r = \sqrt{(-6)^{2}+(6\sqrt{3} )^{2}}\\r = \sqrt{36+(36*3)} \\r = \sqrt{144}\\ r = 12

For the argument;

\theta = tan^{-1} \frac{6\sqrt{3} }{-6} \\\theta = tan^{-1}-\sqrt{3}  \\\theta = -60^{0}

Since tan is negative in the 2nd and 4th quadrant, in the 2nd quadrant,

\theta =180-60\\\theta = 120^{0}

z = 12(cos120°+isin120°)

This gives the required expression.

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