Find the standard form of the complex number $5 \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right)$ .Round values to the nearest hundredth.
A) $- 1.29 - 4.83 i$
B) $2.5 + 4.33 i$
C) $1.29 + 4.83 i$
D) $1.29 - 4.83 i$
E) $- 2.5 - 4.33 i$

Question

Quizplus · Accepted Answer

The standard form of a complex number is $a + bi$, where $a$ and $b$ are the real and imaginary parts, respectively. The given complex number is in polar form, $r(\cos \theta + i\sin \theta)$, where $r = 5$ and $\theta = \frac{5\pi}{12}$. To convert it to standard form, we calculate $a = r\cos\theta$ and $b = r\sin\theta$. Using a calculator, $\cos\frac{5\pi}{12} \approx 0.2588$ and $\sin\frac{5\pi}{12} \approx 0.9659$. Thus, $a = 5 \times 0.2588 \approx 1.29$ and $b = 5 \times 0.9659 \approx 4.83$, making the standard form $1.29 + 4.83i$.

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Find the Standard Form of the Complex Number $5 \left( \cos \frac { 5 \pi } { 12 } + i \sin \frac { 5 \pi } { 12 } \right)$