How can I express the complex number in the form a + bi using cos 315° and sin 315°?

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To express a complex number in the form of a + bi, where i is the imaginary unit, we start with the polar representation of complex numbers. The polar form of a complex number is given by:

r(cos θ + i sin θ)

Here, r is the magnitude and θ is the angle in degrees or radians. For the angle 315°, we can find both cos 315° and sin 315° to convert the representation.

Step 1: Calculate cos 315° and sin 315°

From trigonometric values:

  • cos 315° = cos(360° – 45°) = cos 45° = √2 / 2
  • sin 315° = sin(360° – 45°) = -sin 45° = –√2 / 2

Step 2: Substitute in the Polar Form

Substituting the values we found into the polar form gives:

r(cos 315° + i sin 315°) = r(√2 / 2 – i√2 / 2)

Step 3: Choosing an Appropriate Magnitude

For this example, let’s choose a unit magnitude, or r = 1. Therefore:

Substituting r = 1 gives:

1(√2 / 2 – i√2 / 2) = √2 / 2 – i√2 / 2

Final Result

Thus, the complex number in the form a + bi is:

a = √2 / 2 and b = -√2 / 2

So, the final expression is:

√2 / 2 – i√2 / 2

This can also be interpreted in terms of its polar properties, illustrating how angles and magnitudes can represent complex numbers in various forms.

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