Question

If \[ z=\cos\theta+i\sin\theta, \] then \(|z|\) is equal to:

• A. \(0\)
• B. \(1\)
• C. \(\sqrt2\)
• D. \(\sqrt3\)

Hint:

Any complex number of the form \[ \cos\theta+i\sin\theta \] always has modulus \(1\) because it lies on the unit circle.

Answer 1

The Correct Option is B

Concept: For any complex number \[ z=a+ib, \] the modulus is defined as \[ |z| = \sqrt{a^2+b^2}. \] Geometrically, the modulus represents the distance of the corresponding point from the origin in the Argand plane. The expression \[ z=\cos\theta+i\sin\theta \] is known as the trigonometric form of a complex number and lies on the unit circle.

Step 1: Compare with the standard form. The given complex number is \[ z=\cos\theta+i\sin\theta. \] Thus \[ a=\cos\theta \] and \[ b=\sin\theta. \]

Step 2: Apply the modulus formula. \[ |z| = \sqrt{\cos^2\theta+\sin^2\theta}. \]

Step 3: Use the fundamental trigonometric identity. We know that \[ \sin^2\theta+\cos^2\theta=1. \] Therefore, \[ |z| = \sqrt{1}. \]

Step 4: Simplify. \[ |z| = 1. \]

Step 5: Geometric verification. The point \[ (\cos\theta,\sin\theta) \] lies on the unit circle. Hence its distance from the origin must be \[ 1. \] This confirms the result.

Step 6: Final Conclusion. \[ \boxed{|z|=1} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]