How do you write the complex number in the form a + bi when given the expression (5/2) * (cos 150° + i sin 150°)?

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To express the complex number
(5/2) * (cos(150°) + i sin(150°)) in the form a + bi, we first need to evaluate the trigonometric functions involved.

The angle 150° is located in the second quadrant of the unit circle. Its cosine and sine values are:

  • cos(150°) = -√3/2
  • sin(150°) = 1/2

Next, we can substitute these values into our expression:

We start with:
(5/2) * (cos(150°) + i sin(150°))

Substituting the trigonometric values:
(5/2) * (-√3/2 + i(1/2))

Now, distribute
(5/2):

  • Real part: (5/2) * (-√3/2) = -5√3/4
  • Imaginary part: (5/2) * (1/2) = 5/4

This gives us the complex number in the required form
a + bi:

-5√3/4 + (5/4)i

Thus, the complex number in rectangular form is:

-5√3/4 + (5/4)i

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