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A complex number z_(1) has a magnitude |z_(1)|=4 and an angle theta_(1)=330^(@). Express z_(1) in rectangular form, as z_(1)=a+bi. Express a+bi in exact terms. z_(1)=◻+=^(+x)

A complex number z1 z_{1} has a magnitude ∣z1∣=4 \left|z_{1}\right|=4 and an angle θ1=330∘ \theta_{1}=330^{\circ} .\newlineExpress z1 z_{1} in rectangular form, as z1=a+bi z_{1}=a+b i .\newlineExpress a+bi a+b i in exact terms.\newlinez1=□ z_{1}=\square

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Q. A complex number z1 z_{1} has a magnitude ∣z1∣=4 \left|z_{1}\right|=4 and an angle θ1=330∘ \theta_{1}=330^{\circ} .\newlineExpress z1 z_{1} in rectangular form, as z1=a+bi z_{1}=a+b i .\newlineExpress a+bi a+b i in exact terms.\newlinez1=□ z_{1}=\square
  1. Conversion formula: To express a complex number in rectangular form, we use the polar to rectangular conversion formula, which is z=r(cos⁡(θ)+isin⁡(θ))z = r(\cos(\theta) + i\sin(\theta)), where rr is the magnitude and θ\theta is the angle in radians.
  2. Convert angle to radians: First, we need to convert the angle from degrees to radians because the trigonometric functions in the formula require the angle to be in radians. The conversion is done by multiplying the degree measure by π/180\pi/180. \newlineθ1\theta_{1} in radians = 330×(π/180)=(11/6)π330 \times (\pi/180) = (11/6)\pi
  3. Calculate cosine and sine: Now we can calculate the cosine and sine of θ1\theta_{1} in radians.\newlinecos⁡(θ1)=cos⁡(116π)\cos(\theta_{1}) = \cos\left(\frac{11}{6}\pi\right) and sin⁡(θ1)=sin⁡(116π)\sin(\theta_{1}) = \sin\left(\frac{11}{6}\pi\right)
  4. Substitute values into formula: Using the unit circle or trigonometric identities, we know that cos⁡(116π)=cos⁡(π−16π)=cos⁡(56π)=−cos⁡(16π)\cos\left(\frac{11}{6}\pi\right) = \cos\left(\pi - \frac{1}{6}\pi\right) = \cos\left(\frac{5}{6}\pi\right) = -\cos\left(\frac{1}{6}\pi\right) and sin⁡(116π)=−sin⁡(16π)\sin\left(\frac{11}{6}\pi\right) = -\sin\left(\frac{1}{6}\pi\right), since 116π\frac{11}{6}\pi is in the fourth quadrant where cosine is positive and sine is negative.
  5. Simplify expression: The exact values for cos⁡(16π)\cos\left(\frac{1}{6}\pi\right) and sin⁡(16π)\sin\left(\frac{1}{6}\pi\right) are 3/2\sqrt{3}/2 and 1/21/2, respectively. Therefore, cos⁡(116π)\cos\left(\frac{11}{6}\pi\right) = −3/2-\sqrt{3}/2 and sin⁡(116π)\sin\left(\frac{11}{6}\pi\right) = −1/2-1/2.
  6. Simplify expression: The exact values for cos⁡(16π)\cos\left(\frac{1}{6}\pi\right) and sin⁡(16π)\sin\left(\frac{1}{6}\pi\right) are 3/2\sqrt{3}/2 and 1/21/2, respectively. Therefore, cos⁡(116π)=−3/2\cos\left(\frac{11}{6}\pi\right) = -\sqrt{3}/2 and sin⁡(116π)=−1/2\sin\left(\frac{11}{6}\pi\right) = -1/2.Now we can substitute the values of rr, cos⁡(θ1)\cos(\theta_{1}), and sin⁡(θ1)\sin(\theta_{1}) into the formula to get the rectangular form of z1z_{1}.sin⁡(16π)\sin\left(\frac{1}{6}\pi\right)00
  7. Simplify expression: The exact values for cos⁡(16π)\cos\left(\frac{1}{6}\pi\right) and sin⁡(16π)\sin\left(\frac{1}{6}\pi\right) are 3/2\sqrt{3}/2 and 1/21/2, respectively. Therefore, cos⁡(116π)=−3/2\cos\left(\frac{11}{6}\pi\right) = -\sqrt{3}/2 and sin⁡(116π)=−1/2\sin\left(\frac{11}{6}\pi\right) = -1/2.Now we can substitute the values of rr, cos⁡(θ1)\cos(\theta_{1}), and sin⁡(θ1)\sin(\theta_{1}) into the formula to get the rectangular form of z1z_{1}.\newlinesin⁡(16π)\sin\left(\frac{1}{6}\pi\right)00Simplify the expression to get the final rectangular form of z1z_{1}.\newlinesin⁡(16π)\sin\left(\frac{1}{6}\pi\right)22

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