Write the complex number 100(cos240°+ i sin240°) in standard form.
100(-.5 -.866i)
- 50 - 50 sqrt3 i
To write a complex number in standard form, we need to express it in terms of the real and imaginary parts.
Given that the complex number is 100(cos240° + i sin240°), we can rewrite this using Euler's formula, which states that e^ix = cos(x) + i sin(x).
Using Euler's formula, we can rewrite the complex number as:
100 * e^(i * 240°)
Now, using the exponential form of a complex number, which is:
z = r * e^(i * theta)
where r is the magnitude of the complex number and theta is the argument (angle in radians), we can express the complex number as:
z = 100 * e^(i * (240° * (pi/180)))
Converting degrees to radians, we get:
z = 100 * e^(i * (240 * (pi/180)))
Now, calculating the exponential expression:
z = 100 * e^(i * (240 * (pi/180)))
= 100 * e^(i * (4/3)π)
Finally, since e^(i * (4/3)π) is a complex number on the unit circle, we can express it in standard form as:
z = 100 * cos(4/3π) + 100 * i * sin(4/3π)
Simplifying further, we have:
z = 100 * (-1/2) + 100i * (√3/2)
= -50 + 50√3i
Therefore, the complex number 100(cos240° + i sin240°) can be written in standard form as -50 + 50√3i.