Change the complex number `4(cos 300^(@) + i sin 300^(@))` to cartesian form

Answer

To convert the complex number \( 4(\cos 300^\circ + i \sin 300^\circ) \) to Cartesian form, we can follow these steps:

### Step 1: Rewrite the angle
We can express \( 300^\circ \) in terms of \( 360^\circ \):
\[ 300^\circ = 360^\circ - 60^\circ \]
This helps us to use the properties of trigonometric functions.

### Step 2: Apply the cosine and sine properties
Using the properties of cosine and sine:
- \( \cos(360^\circ - \theta) = \cos(\theta) \)
- \( \sin(360^\circ - \theta) = -\sin(\theta) \)

We can rewrite the complex number:
\[ \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ \]
\[ \sin 300^\circ = \sin(360^\circ - 60^\circ) = -\sin 60^\circ \]

### Step 3: Substitute the values
Now substituting these values into our expression:
\[ 4(\cos 300^\circ + i \sin 300^\circ) = 4\left(\cos 60^\circ - i \sin 60^\circ\right) \]

### Step 4: Calculate cosine and sine values
We know:
\[ \cos 60^\circ = \frac{1}{2} \quad \text{and} \quad \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Substituting these values gives:
\[ 4\left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right) \]

### Step 5: Simplify the expression
Distributing the 4:
\[ = 4 \cdot \frac{1}{2} - 4 \cdot i \cdot \frac{\sqrt{3}}{2} \]
\[ = 2 - 2i\sqrt{3} \]

### Final Answer
Thus, the Cartesian form of the complex number is:
\[ 2 - 2\sqrt{3}i \]

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