Convert the complex number from polar to rectangular form.
z = 8 cis(240°)
Please leave in exact form, no decimals
The Correct Answer and Explanation is :
To convert the complex number ( z = 8 \text{ cis}(240^\circ) ) from polar to rectangular form, we first need to understand what the polar form and the ( \text{cis} ) notation represent. The polar form of a complex number is expressed as:
[
z = r \text{ cis}(\theta) = r(\cos(\theta) + i\sin(\theta))
]
where:
- ( r ) is the modulus (or magnitude) of the complex number,
- ( \theta ) is the argument (or angle) in degrees, and
- ( i ) is the imaginary unit.
In this case, ( r = 8 ) and ( \theta = 240^\circ ).
Next, we calculate the cosine and sine of ( 240^\circ ):
- Finding ( \cos(240^\circ) ) and ( \sin(240^\circ) ):
- ( 240^\circ ) lies in the third quadrant of the unit circle, where both cosine and sine values are negative.
- The reference angle for ( 240^\circ ) is ( 240^\circ – 180^\circ = 60^\circ ). Using known values:
- ( \cos(60^\circ) = \frac{1}{2} )
- ( \sin(60^\circ) = \frac{\sqrt{3}}{2} ) Therefore:
- ( \cos(240^\circ) = -\cos(60^\circ) = -\frac{1}{2} )
- ( \sin(240^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2} )
- Substituting these values back into the rectangular form:
[
z = 8 \left( \cos(240^\circ) + i\sin(240^\circ) \right)
= 8 \left( -\frac{1}{2} + i \left(-\frac{\sqrt{3}}{2}\right) \right)
] Now, multiplying through by 8 gives:
[
z = 8 \left( -\frac{1}{2} \right) + 8 \left( -\frac{\sqrt{3}}{2} \right) i
= -4 – 4\sqrt{3}i
]
Thus, the rectangular form of the complex number ( z = 8 \text{ cis}(240^\circ) ) is:
[
\boxed{-4 – 4\sqrt{3}i}
]
In summary, the conversion from polar to rectangular form involves calculating the cosine and sine of the angle, then using these values to express the complex number in terms of its real and imaginary components. The final rectangular form can be useful in various applications, including electrical engineering and complex analysis, where operations on complex numbers are performed more conveniently in rectangular coordinates.