AI-generated answer
(a) To write the number e⁻ᶦΠ/³ in the form a + bi, we can use Euler's formula, which states that e^(ix) = cos(x) + i*sin(x).
Here, we have e⁻ᶦΠ/³. To determine the values of a and b, we need to express the exponent in terms of its cosine and sine components.
Using Euler's formula, we can rewrite e⁻ᶦΠ/³ as cos(-Π/³) + i*sin(-Π/³).
Simplifying further, we have cos(-Π/³) - i*sin(Π/³).
Therefore, the number e⁻ᶦΠ/³ can be expressed in the form a + bi as cos(-Π/³) - i*sin(Π/³).
(b) Similarly, to write e⁽¹⁺ᶦ²Π⁾ / e⁽⁻¹⁺ᶦΠ/²⁾ in the form a + bi, we can apply Euler's formula.
Using Euler's formula, we can express e⁽¹⁺ᶦ²Π⁾ as cos(1+2Π) + i*sin(1+2Π), and e⁽⁻¹⁺ᶦΠ/²⁾ as cos(-1+Π/2) + i*sin(-1+Π/2).
Dividing the two expressions, we get (cos(1+2Π) + i*sin(1+2Π)) / (cos(-1+Π/2) + i*sin(-1+Π/2)).
Simplifying this further will yield a + bi form, but the specific values of a and b depend on the calculations.
(c) Lastly, to write eᵉᶦ in the form a + bi, we can use Euler's formula again.
Applying Euler's formula, we have e^(e*i) = cos(e) + i*sin(e).
Therefore, the number eᵉᶦ can be expressed in the form a + bi as cos(e) + i*sin(e).
It's important to note that without specific values for e in each equation, we cannot provide the exact numerical values for a and b. However, the expressions cos(x) and sin(x) represent the real and imaginary parts of a complex number, respectively, allowing us to write the given numbers in the form a + bi.