Question

Write the expression in the standard form a + b i.

[4(cos 100° + i sin 100°)]³ = ____
(Simplify your answer, including any radicals.

Answer 1

Using De Moivre's Theorem and trigonometric identities, the expression 4(cos 100° + i sin 100°)³ can be written in the standard form a + b i by calculating cos(300°) and sin(300°) and then multiplying by 4.

Step-by-step explanation:

The student's question involves trigonometry and complex numbers, where they are asked to write the expression 4(cos 100° + i sin 100°)³ in the standard form a + b i, which simplifies to a complex number in rectangular coordinates. To achieve this, we must apply De Moivre's Theorem, which states that (cos θ + i sin θ)ⁿᵗ = cos(nθ) + i sin(nθ) for any integer n. First, we find the trigonometric expressions for cos(300°) and sin(300°), since 100° is the angle given and we are cubing the expression (multiplying the angle by 3). After calculating these values, we will then multiply the result by 4, the coefficient given in the original expression, to find the standard form a + b i.

To demonstrate, let's carry out the steps:

1. Use De Moivre's Theorem to find the trigonometric expressions for cos(300°) and sin(300°).
2. Multiply these values by 4 to get the standard form.
3. Simplify the expression, which will involve using trigonometric identities as necessary.

The correct application of this process will yield the complex number in standard form as the final answer.