Question

Express the given complex number (-3) in the polar form.​

Answer 1

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Express the given complex number (-3) in the polar form.

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➡️Given complex number is -3

➡️Let r cos θ = -3 …(1)

and r sinθ = 0 …(2)

➡️Squaring and adding (1) and (2), we get

➡️r^2cos^2θ +r^2sin^2θ = (-3)^2

➡️Take r^2 outside from L.H.S, we get

➡️r^2(cos^2θ+sin^2θ) = 9

➡️We know that, cos^2θ+sin^2θ = 1, then the above equation becomes

➡️r^2 = 9

➡️r = 3 (Conventionally, r > 0)

➡️Now, subsbtitute the value of r in (1) and (2)

➡️3 cos θ = -3 and 3 sinθ = 0

➡️Cos θ = -1 and sinθ = 0

➡️Therefore, θ = π

➡️Hence, the polar representation is,

➡️-3 = rcos θ + i rsin θ

➡️3cosπ+ 3sin π = 3(cosπ+isin π)

➡️Thus, the required polar form is 3cosπ+ i3sin π = 3(cosπ+isin π)

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Answer 2

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Given, complex number is -3.

Let r cos θ = -3 …(1)

and r sin θ = 0 …(2)

Squaring and adding (1) and (2), we get

r^2cos^2θ + r^2sin^2θ = (-3)^2

Take r^2 outside from L.H.S, we get

r^2(cos^2θ + sin^2θ) = 9

We know that, cos^2θ + sin^2θ = 1, then the above equation becomes,

r^2 = 9

r = 3 (Conventionally, r > 0)

Now, subsbtitute the value of r in (1) and (2)

3 cos θ = -3 and 3 sin θ = 0

cos θ = -1 and sin θ = 0

Therefore, θ = π

Hence, the polar representation is,

-3 = r cos θ + i r sin θ

3 cos π + 3 sin π = 3(cos π + i sin π)

Thus, the required polar form is 3 cos π+ 3i sin π = 3(cos π+i sin π)

Hope it's Helpful....:)