Final answer:
To evaluate [2(cos 75° + i sin 75°)]⁵, we use De Moivre's Theorem. The expression simplifies to (32)(cos(15°) + i sin(15°)), and requires the cosine and sine of 15° multiplied by 32 to find the 'a + bi' form.
Step-by-step explanation:
To evaluate and express the given expression [2(cos 75° + i sin 75°)]⁵ in 'a + bi' form, we can use De Moivre's Theorem. This theorem states that for any complex number in polar form (r(cos θ + i sin θ)), its power n can be found as (r^n)(cos(nθ) + i sin(nθ)).
Starting with the given expression, we first notice that 2 is the modulus (r) and 75° is the angle (θ). So, according to De Moivre's Theorem our expression becomes:
(2^5)(cos(5×75°) + i sin(5×75°))
(32)(cos(375°) + i sin(375°))
Since 375° is greater than 360°, we can subtract 360° to simplify the angle (375° - 360° = 15°). This leaves us with:
(32)(cos(15°) + i sin(15°))
We then find the cosine and sine of 15° and multiply by 32 to get the final answer in 'a + bi' form.