#### 78.54 cm²Question: Compute the minimum value of $ |\cos x + i\sin x + 1| $ as $ x $ ranges over all real numbers.
["Compute the Minimum Value of ( |\cos x + i\sin x + 1| ) for Real ( x )", "In complex analysis, expressions involving ( \cos x + i\sin x ) often appear in trigonometric and geometric interpretations. The expression ( |\cos x + i\sin x + 1| ) involves the modulus (or absolute value) of a complex number, and understanding its minimum value over all real ( x ) reveals important properties of the unit circle confined in the complex plane.", "---", "### Step 1: Recognize the complex form", "Recall Euler’s formula:", "[\n\cos x + i\sin x = e^{ix}\n]", "So the expression becomes:", "[\n|e^{ix} + 1| = |\cos x + i\sin x + 1|\n]", "We seek the minimum of this modulus as ( x \in \mathbb{R} ).", "---", "### Step 2: Rewrite using geometry", "The complex number ( \cos x + i\sin x ) lies on the unit circle centered at the origin:", "[\nz = \cos x + i\sin x \quad \ ext{with } |z| = 1\n]", "Then:", "[\n|\cos x + i\sin x + 1| = |z + 1|\n]", "This is the distance from the point ( z = \cos x + i\sin x ) (a point on the unit circle) to the fixed point ( -1 ) on the real axis.", "Geometrically, we are finding the minimum distance from any point on the unit circle to the point ( (-1, 0) ) on the real axis.", "---", "### Step 3: Use vector geometry", "Let ( z = (\cos x, \sin x) ) and we compute:", "[\n|z + 1| = \sqrt{(\cos x + 1)^2 + (\sin x)^2}\n]", "Expand:", "[\n(\cos x + 1)^2 + \sin^2 x = \cos^2 x + 2\cos x + 1 + \sin^2 x\n]", "Using ( \cos^2 x + \sin^2 x = 1 ):", "[\n= 1 + 2\cos x + 1 = 2 + 2\cos x\n]", "So:", "[\n|z + 1| = \sqrt{2 + 2\cos x} = \sqrt{2(1 + \cos x)}\n]", "---", "### Step 4: Minimize the expression", "We now minimize:", "[\nf(x) = \sqrt{2(1 + \cos x)}\n]", "Since the square root is increasing, minimizing ( f(x) ) is equivalent to minimizing ( 1 + \cos x ).", "The minimum value of ( \cos x ) is ( -1 ), achieved when ( x = \pi + 2k\pi ), ( k \in \mathbb{Z} ).", "Then:", "[\n1 + \cos x = 0 \quad \Rightarrow \quad f(x) = \sqrt{2 \cdot 0} = 0\n]", "But wait — is this possible?", "Check:", "At ( x = \pi ),\n( \cos\pi = -1 ), ( \sin\pi = 0 ), so", "[\nz + 1 = (-1 + i \cdot 0) + 1 = 0\n]", "Thus,", "[\n|z + 1| = |0| = 0\n]", "This confirms the minimum is 0, achieved when ( x = \pi ), ( 3\pi ), etc.", "---", "### Step 5: Conclusion", "The minimum value of ( |\cos x + i\sin x + 1| ) over all real ( x ) is:", "[\n\boxed{0}\n]", "This result highlights a key geometric fact: the origin ( 0 ) lies on the unit circle, so the complex number ( -1 ) (shifted by +1) is exactly at a distance zero from the point on the unit circle passing through it.", "---", "### SEO Keywords Used:\n\cos x, i sin x, |cos x + i sin x + 1|, complex number modulus, unit circle distance, minimum modulus, geometry of complex numbers, trigonometric identity, minimum value, mathematical computation, Euler’s formula", "---", "Summary: The minimum value of ( |\cos x + i\sin x + 1| ) is ( 0 ), achieved when ( x = \pi + 2k\pi ), because ( \cos x = -1 ) places the point directly opposite the origin, aligning ( z + 1 = 0 )."]
