How do you express the complex number 6cos(330°) + i sin(330°) in the standard form a + bi?

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To express the complex number given as 6cos(330°) + i sin(330°) in the standard form a + bi, we first need to calculate the values of cos(330°) and sin(330°).

Step 1: Calculate cos(330°)

The angle 330° is in the fourth quadrant of the unit circle. In this quadrant, cosine values are positive. The reference angle for 330° can be found by subtracting it from 360°:

360° – 330° = 30°

Thus, we have:

cos(330°) = cos(30°) = √3 / 2

Step 2: Calculate sin(330°)

For the sine value, in the fourth quadrant, sine values are negative. Therefore:

sin(330°) = -sin(30°) = -1 / 2

Step 3: Substitute the values

Now substituting these values back into the complex number, we get:

6cos(330°) + i sin(330°) = 6(√3 / 2) + i(-1 / 2)

Step 4: Simplify the expression

This simplifies to:

6(√3 / 2) = 3√3 and i(-1 / 2) = – (1 / 2)i

So, the complex number can be expressed as:

3√3 – (1 / 2)i

Conclusion

In standard form, the complex number 6cos(330°) + i sin(330°) can be written as:

3√3 – (1 / 2)i

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