Use a calculator to help write the complex number in standard form. Round the numbers in your answer to the nearest hundredth. 1 cis 200°
The Correct Answer and Explanation is :
o express the complex number (1 \, \text{cis} \, 200^\circ) in standard form, we first understand that the notation ( \text{cis} \, \theta ) stands for ( \cos \theta + i \sin \theta ), where (i) is the imaginary unit.herefore, (1 \, \text{cis} \, 200^\circ) translates to ( \cos 200^\circ + i \sin 200^\circ ).
Calculating ( \cos 200^\circ ) and ( \sin 200^\circ ):
- Cosine Component:
- sing a calculator, ( \cos 200^\circ ) yields approximately ( -0.94 ) when rounded to the nearest hundredth.
- Sine Component:
- imilarly, ( \sin 200^\circ ) results in approximately ( -0.34 ) when rounded to the nearest hundredth.
hus, (1 \, \text{cis} \, 200^\circ) in standard form is:
[ -0.94 – 0.34i ]
Explanation:
omplex numbers can be represented in two primary forms: standard (rectangular) form and polar form.he standard form is ( a + bi ), where (a) is the real part and (b) is the imaginary part.he polar form expresses a complex number in terms of its magnitude (or modulus) and angle (or argument) as ( r \, \text{cis} \, \theta ), where ( r ) is the magnitude and ( \theta ) is the angle in degrees or radians.
he conversion between these forms utilizes trigonometric functions:
- ( a = r \cos \theta )- ( b = r \sin \theta )
n this problem, the magnitude ( r ) is 1, and the angle ( \theta ) is ( 200^\circ ).y calculating ( \cos 200^\circ ) and ( \sin 200^\circ ), we determine the real and imaginary components of the complex number.
ngles in trigonometry are often measured from the positive x-axis, with positive angles indicating counterclockwise rotation.n angle of ( 200^\circ ) places the terminal side in the third quadrant of the unit circle, where both cosine and sine values are negative.his aligns with our calculated values of ( -0.94 ) for ( \cos 200^\circ ) and ( -0.34 ) for ( \sin 200^\circ ).
y substituting these values into the standard form ( a + bi ), we obtain ( -0.94 – 0.34i ).his representation provides a clear view of the complex number’s real and imaginary parts, which is often more intuitive for addition, subtraction, and other algebraic operations.
n summary, converting a complex number from polar to standard form involves evaluating the cosine and sine of the given angle to find the real and imaginary components, respectively.his process is fundamental in complex number arithmetic and has applications across various fields, including engineering, physics, and applied mathematics.