Question

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Complete each statement given w = 2(cos(90°) + i sin (90°)) and z= √ (cos(250) + i sin(225°)).

Answer 1

To complete each statement, we will need to perform some operations on the given complex numbers w and z.

1. Find w^3:

We can use De Moivre's theorem to raise w to the third power:

w^3 = [2(cos(90°) + i sin(90°))]^3

= 2^3(cos(90°3) + i sin(90°3))

= 8(cos(270°) + i sin(270°))

Therefore, w^3 = 8(cos(270°) + i sin(270°)).

2. Simplify z^2:

To simplify z^2, we can use the identity cos(2θ) = 2cos^2(θ) - 1 to simplify the cosine term:

z^2 = [√(cos(250°) + i sin(225°))]^2

= cos(2250°) + i sin(2225°)

= cos(500°) + i sin(450°)

= cos(140°) - i sin(90°)

Therefore, z^2 = cos(140°) - i sin(90°).

Note: We can also simplify the square root of the cosine term using the identity cos(2θ) = 1 - 2sin^2(θ), but this would result in a more complicated expression for z^2.

3. Find the product wz:

We can simply multiply w and z using the distributive property:

wz = 2(cos(90°) + i sin(90°)) * √(cos(250°) + i sin(225°))

= 2√(cos(90°)cos(250°) - sin(90°)sin(250°) + i(cos(90°)sin(250°) + sin(90°)cos(250°)))

= 2√(-sin(250°) + i cos(250°))

Therefore, wz = 2√(-sin(250°) + i cos(250°)).

Answer 2

Given
`\( w = 2(\cos(90^\circ) + i \sin(90^\circ)) \) and \( z = √(\cos(250^\circ) + i \sin(225^\circ)) \)`, let's evaluate each expression:

1. For
`\( w \)`:

Applying Euler's formula
`\( e^(i\theta) = \cos(\theta) + i \sin(\theta) \)`, we find that
`\( w = 2i \)`.

2. For
`\( z \)`:

Using Euler's formula and recognizing
`\( √(i) = e^(i(\pi/4)) \),` we have
`\( z = √(-i) \)`. For
`\(250^\circ\)`, the cosine is
`\(-√(2)/2\)` and the sine is
`\(-1/√(2)\)`, leading to
`\( z = √(-i) \).`

Step-by-step explanation:

1. Simplifying
`\( w \)`:

The expression
`\( w = 2(\cos(90^\circ) + i \sin(90^\circ)) \)` can be simplified using Euler's formula, which relates complex exponentials to trigonometric functions. For
`\(90^\circ\)`, the cosine is 0, and the sine is 1. Thus,
`\( w = 2i \)`.

2. Simplifying
`\( z \)`:

The expression
`\( z = √(\cos(250^\circ) + i \sin(225^\circ)) \)` involves the square root of a complex number. Using Euler's formula and recognizing
`\( √(i) = e^(i(\pi/4)) \), we evaluate \( z = √(-i) \). For \(250^\circ\)`, the cosine is
`\(-√(2)/2\)` and the sine is
`\(-1/√(2)\).` Therefore,
`\( z = √(-i) \).`

In conclusion, the calculations involve applying Euler's formula to express complex numbers in trigonometric form and recognizing standard trigonometric values for the specified angles. The final results are
`\( w = 2i \) and \( z = √(-i) \).`