Questions: Find standard notation, a+bi. 18(cos(150°)+i sin(150°))

Transcript text: Find standard notation, $a+b i$. \[ 18\left(\cos \left(150^{\circ}\right)+i \sin \left(150^{\circ}\right)\right) \]

Solution

Solution Steps

To convert the given expression from polar form to standard notation $a + bi$, we need to use the formulas $a = r \cos(\theta)$ and $b = r \sin(\theta)$, where $r$ is the magnitude and $\theta$ is the angle in degrees. Here, $r = 18$ and $\theta = 150^\circ$.

Step 1: Convert Angle to Radians

To convert the angle from degrees to radians, we use the conversion formula: \[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \] For $\theta = 150^\circ$: \[ \theta_{\text{radians}} = 150 \times \frac{\pi}{180} = 2.6179938779914944 \]

Step 2: Calculate $a$ and $b$

Using the formulas for $a$ and $b$: \[ a = r \cos(\theta) \quad \text{and} \quad b = r \sin(\theta) \] Substituting $r = 18$ and $\theta = 2.6179938779914944$: \[ a = 18 \cos(2.6179938779914944) \approx -15.5885 \] \[ b = 18 \sin(2.6179938779914944) \approx 9.0000 \]

Step 3: Write in Standard Notation

Combining the values of $a$ and $b$, we express the result in standard notation: \[ a + bi = -15.5885 + 9.0000i \]