Solved on Nov 29, 2023

Rewrite complex number $10(\cos 240^\circ + i \sin 240^\circ)$ in rectangular form.

STEP 1

Assumptions
1. The complex number is in polar form: $10\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$
2. We need to convert this to rectangular form. The rectangular form of a complex number is $a + bi$ , where $a$ is the real part and $b$ is the imaginary part.

STEP 2

The rectangular form of a complex number in polar form $r(\cos \theta + i \sin \theta)$ is given by $r\cos \theta + r i \sin \theta$ .
So we need to calculate $r\cos \theta$ and $r i \sin \theta$ separately.

STEP 3

First, let's calculate $r\cos \theta$ which will give us the real part of the complex number.
$Real\, part = r\cos \theta$

STEP 4

Now, plug in the given values for $r$ and $\theta$ to calculate the real part.
$Real\, part = 10\cos 240^{\circ}$

STEP 5

Calculate the real part.
$Real\, part = 10\cos 240^{\circ} = -5$

STEP 6

Next, let's calculate $r i \sin \theta$ which will give us the imaginary part of the complex number.
$Imaginary\, part = r i \sin \theta$

STEP 7

Now, plug in the given values for $r$ and $\theta$ to calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ}$

STEP 8

Calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ} = -10i\sqrt{3}$

STEP 9

Finally, we can write the complex number in rectangular form by combining the real part and the imaginary part.
$Rectangular\, form = Real\, part + Imaginary\, part$

STEP 10

Plug in the values for the real part and the imaginary part to write the complex number in rectangular form.
$Rectangular\, form = -5 - 10i\sqrt{3}$
The rectangular form of $10\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$ is $-5 - 10i\sqrt{3}$ .

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Rewrite complex number $10(\cos 240^\circ + i \sin 240^\circ)$ in rectangular form.

STEP 1

Assumptions
1. The complex number is in polar form: $10\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$
2. We need to convert this to rectangular form. The rectangular form of a complex number is $a + bi$ , where $a$ is the real part and $b$ is the imaginary part.

STEP 2

The rectangular form of a complex number in polar form $r(\cos \theta + i \sin \theta)$ is given by $r\cos \theta + r i \sin \theta$ .
So we need to calculate $r\cos \theta$ and $r i \sin \theta$ separately.

STEP 3

First, let's calculate $r\cos \theta$ which will give us the real part of the complex number.
$Real\, part = r\cos \theta$

STEP 4

Now, plug in the given values for $r$ and $\theta$ to calculate the real part.
$Real\, part = 10\cos 240^{\circ}$

STEP 5

Calculate the real part.
$Real\, part = 10\cos 240^{\circ} = -5$

STEP 6

Next, let's calculate $r i \sin \theta$ which will give us the imaginary part of the complex number.
$Imaginary\, part = r i \sin \theta$

STEP 7

Now, plug in the given values for $r$ and $\theta$ to calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ}$

STEP 8

Calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ} = -10i\sqrt{3}$

STEP 9

Finally, we can write the complex number in rectangular form by combining the real part and the imaginary part.
$Rectangular\, form = Real\, part + Imaginary\, part$

STEP 10

Plug in the values for the real part and the imaginary part to write the complex number in rectangular form.
$Rectangular\, form = -5 - 10i\sqrt{3}$
The rectangular form of $10\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)$ is $-5 - 10i\sqrt{3}$ .

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Rewrite complex number 10(cos⁡240∘+isin⁡240∘)10(\cos 240^\circ + i \sin 240^\circ)10(cos240∘+isin240∘) in rectangular form.

STEP 1

STEP 2

STEP 3

First, let's calculate rcos⁡θr\cos \thetarcosθ which will give us the real part of the complex number.Real part=rcos⁡θReal\, part = r\cos \thetaRealpart=rcosθ

STEP 4

Now, plug in the given values for rrr and θ\thetaθ to calculate the real part.Real part=10cos⁡240∘Real\, part = 10\cos 240^{\circ}Realpart=10cos240∘

STEP 5

Calculate the real part.Real part=10cos⁡240∘=−5Real\, part = 10\cos 240^{\circ} = -5Realpart=10cos240∘=−5

STEP 6

Next, let's calculate risin⁡θr i \sin \thetarisinθ which will give us the imaginary part of the complex number.Imaginary part=risin⁡θImaginary\, part = r i \sin \thetaImaginarypart=risinθ

STEP 7

Now, plug in the given values for rrr and θ\thetaθ to calculate the imaginary part.Imaginary part=10isin⁡240∘Imaginary\, part = 10i \sin 240^{\circ}Imaginarypart=10isin240∘

STEP 8

Calculate the imaginary part.Imaginary part=10isin⁡240∘=−10i3Imaginary\, part = 10i \sin 240^{\circ} = -10i\sqrt{3}Imaginarypart=10isin240∘=−10i3​

STEP 9

Finally, we can write the complex number in rectangular form by combining the real part and the imaginary part.Rectangular form=Real part+Imaginary partRectangular\, form = Real\, part + Imaginary\, partRectangularform=Realpart+Imaginarypart

STEP 10

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Rewrite complex number $10(\cos 240^\circ + i \sin 240^\circ)$ in rectangular form.

First, let's calculate $r\cos \theta$ which will give us the real part of the complex number.
$Real\, part = r\cos \theta$

Now, plug in the given values for $r$ and $\theta$ to calculate the real part.
$Real\, part = 10\cos 240^{\circ}$

Calculate the real part.
$Real\, part = 10\cos 240^{\circ} = -5$

Next, let's calculate $r i \sin \theta$ which will give us the imaginary part of the complex number.
$Imaginary\, part = r i \sin \theta$

Now, plug in the given values for $r$ and $\theta$ to calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ}$

Calculate the imaginary part.
$Imaginary\, part = 10i \sin 240^{\circ} = -10i\sqrt{3}$

Finally, we can write the complex number in rectangular form by combining the real part and the imaginary part.
$Rectangular\, form = Real\, part + Imaginary\, part$