Modulus of cos(theta) + i sin(theta)

To find the modulus of the complex number cos(theta) + i sin(theta), we can use the formula for the modulus of a complex number:

|z| = √(a^2 + b^2)

where a and b are the real and imaginary parts of the complex number, respectively.

In this case, the real part of the complex number is cos(theta) and the imaginary part is sin(theta). Therefore, we can rewrite the complex number as:

cos(theta) + i sin(theta)

Now, let's calculate the modulus by substituting the values into the formula.

Calculation:

|z| = √(cos^2(theta) + sin^2(theta))

Using the trigonometric identity cos^2(theta) + sin^2(theta) = 1, we can simplify the expression:

|z| = √(1)

Since the square root of 1 is 1, the modulus of the complex number cos(theta) + i sin(theta) is 1.

Explanation:

The modulus of a complex number represents its distance from the origin in the complex plane. In the case of cos(theta) + i sin(theta), the modulus is 1, which means the complex number lies on the unit circle centered at the origin.

Visually, this complex number represents a point on the unit circle that corresponds to an angle theta in the polar coordinate system. The real part (cos(theta)) gives the x-coordinate of the point, and the imaginary part (sin(theta)) gives the y-coordinate.

By calculating the modulus, we can determine the length or magnitude of the complex number without considering its direction or angle. In this case, the length is always 1, indicating that the complex number is always at a distance of 1 from the origin.

In summary, the modulus of cos(theta) + i sin(theta) is always 1, indicating that it lies on the unit circle centered at the origin in the complex plane.

Root over 1