To express a complex number in the form a + bi, we need to start by using the known values of cosine and sine for the angle 315°. The cosine and sine of 315° can be derived from the unit circle, where 315° is located in the fourth quadrant.
The cosine value is:
cos(315°) = √2 / 2
The sine value is:
sin(315°) = -√2 / 2
Now, we can express the complex number using these values. The standard form of a complex number is usually written as:
z = r(cos θ + i sin θ)
For our case, we will substitute 315° (which is equivalent to -45°) into this formula. The general form is:
z = r(e^(iθ)) = r(cos θ + i sin θ)
If we set r = 1 (unit circle), we can directly write the complex number:
z = cos(315°) + i sin(315°)
Substituting the cosine and sine values, we have:
z = (√2 / 2) + i(-√2 / 2)
Therefore, the complex number in the form a + bi is:
z = (√2 / 2) – i(√2 / 2)
This is how you write a complex number in the a + bi form using the cosine and sine of 315°.