Questions: Simplify: [3(cos 15°+i sin 15°)]^4 A. 81(cos 15°+i sin 15°) B. 81(cos 60°+i sin 60°) C. 45(cos 45°+i sin 45°) D. 45(cos 30°+i sin 30°) E. 15(cos 81°+i sin 81°)

Transcript text: Simplify: $\left[3\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4}$ A. $81\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)$ B. $81\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)$ C. $45\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)$ D. $45\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)$ E. $15\left(\cos 81^{\circ}+i \sin 81^{\circ}\right)$

Solution

Solution Steps

Step 1: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form $ r(\cos \theta + i \sin \theta) $, raising it to the power of $ n $ gives: \[ [r(\cos \theta + i \sin \theta)]^n = r^n (\cos (n\theta) + i \sin (n\theta)). \] Here, $ r = 3 $, $ \theta = 15^\circ $, and $ n = 4 $.

Step 2: Calculate $ r^n $

Compute $ r^n = 3^4 = 81 $.

Step 3: Calculate $ n\theta $

Compute $ n\theta = 4 \times 15^\circ = 60^\circ $.

Step 4: Write the simplified form

Using De Moivre's Theorem, the expression simplifies to: \[ 81 (\cos 60^\circ + i \sin 60^\circ). \]