What is the trigonometric form of $(- 7 + 8 i)$ $(-7+8i)$ ?

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What is the trigonometric form of $(- 7 + 8 i)$ $(-7+8i)$ ?

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Answer 1 · 2015-12-21T17:45:10

$z = \sqrt{113} (cos θ + i sin θ), θ = π - arctan \frac{8}{7}$ $z = sqrt 113 (cos theta + i sin theta), theta = pi - arctan frac{8}{7}$

Explanation:

We search for $z = - 7 + 8 i = r (cos θ + i sin θ), r = | z |, θ = arg z$ $z = -7 + 8i = r(cos theta + i sin theta), r = |z|, theta = text{arg } z$

$- 7 = r cos θ$ $-7 = r cos theta$

$8 = r sin θ$ $8 = r sin theta$

$7^{2} + 8^{2} = 113 = r^{2}$ $7^2 + 8^2 = 113 = r^2$

$\frac{8}{- 7} = tan θ; \frac{π}{2} < θ < π$ $8/-7 = tan theta ; pi/2 < theta < pi$

$tan (π - θ) = \frac{sin (π - θ)}{cos (π - θ)} = \frac{sin θ}{- cos θ} = - tan θ = \frac{8}{7}$ $tan (pi - theta) = sin (pi - theta) / cos (pi - theta) = sin theta /- cos theta = - tan theta = 8/7$

$θ = π - arctan \frac{8}{7}$ $theta = pi - arctan frac{8}{7}$

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What is the trigonometric form of $(- 7 + 8 i)$ $(-7+8i)$ ?

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What is the trigonometric form of (−7+8i) (-7+8i) ?

1 Answer

Explanation:

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What is the trigonometric form of $(- 7 + 8 i)$ $(-7+8i)$ ?