Question: Among all the roots of \(z^8 + z^4 + 1 = 0\), the maximum imaginary part can be expressed as \(\sin \phi\), where \(0^\circ < \phi < 90^\circ\). Find \(\phi\) in degrees.

["Question: Among all the roots of (z^8 + z^4 + 1 = 0), the maximum imaginary part can be expressed as (\sin \phi), where (0^\circ < \phi < 90^\circ). Find (\phi) in degrees.", "---", "Unlocking the Maximum Imaginary Part of (z^8 + z^4 + 1 = 0)", "When analyzing complex roots of polynomial equations, understanding the location and magnitude of their imaginary components is both mathematically rich and elegantly tied to trigonometric functions. Today, we explore the roots of the equation:", "[\nz^8 + z^4 + 1 = 0\n]", "and determine the angle (\phi) such that the maximum imaginary part among these roots equals (\sin \phi), with (\phi) restricted to (0^\circ < \phi < 90^\circ).", "---", "### Step 1: Simplify the Equation Using Substitution", "Let ( w = z^4 ). Then the equation becomes:", "[\nw^2 + w + 1 = 0\n]", "This is a quadratic equation. Solving using the quadratic formula:", "[\nw = \frac{-1 \pm \sqrt{1^2 - 4(1)(1)}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}\n]", "Thus, the two solutions are complex cube roots of unity (excluding 1):", "[\nw = e^{2\pi i/3} \quad \ ext{and} \quad w = e^{-2\pi i/3} = e^{4\pi i/3} \quad (\ ext{mod } 2\pi)\n]", "So, ( z^4 = e^{2\pi i/3} ) and ( z^4 = e^{4\pi i/3} )", "---", "### Step 2: Solve for ( z ) by Root Extraction", "We now solve for ( z ) by taking 4th roots of each value.", "Recall: To find the 4th roots of a complex number ( re^{i\ heta} ), the solutions are:", "[\nz_k = r^{1/4} , e^{i(\ heta + 2k\pi)/4}, \quad k = 0, 1, 2, 3\n]", "#### Case 1: ( z^4 = e^{2\pi i/3} )", "- Modulus: ( (e^{2\pi i/3})^{1/4} = e^{\pi i/6} )\n- The four roots:", "[\nz = e^{i(\pi/6 + 2k\pi)/4} = e^{i(\pi/24 + k\pi/2)} \quad \ ext{for } k = 0,1,2,3\n]", "Angles (in radians): ( \frac{\pi}{24}, \frac{13\pi}{24}, \frac{25\pi}{24}, \frac{37\pi}{24} )", "Convert to degrees:", "- ( \frac{\pi}{24} = 7.5^\circ )\n- ( \frac{13\pi}{24} = 97.5^\circ )\n- ( \frac{25\pi}{24} = 187.5^\circ )\n- ( \frac{37\pi}{24} = 277.5^\circ )", "Imaginary parts: ( \sin(7.5^\circ), \sin(97.5^\circ), \sin(187.5^\circ), \sin(277.5^\circ) )\nSince ( \sin ) is symmetric and periodic, maximum imaginary part from these corresponds to ( \sin(97.5^\circ) )", "#### Case 2: ( z^4 = e^{4\pi i/3} )", "- Base angle: ( \frac{4\pi}{3} \approx 240^\circ ), so:", "[\nz = e^{i(4\pi/3 + 2k\pi)/4} = e^{i(\pi/3 + k\pi/2)} \quad \ ext{for } k = 0,1,2,3\n]", "Angles: ( 60^\circ, 150^\circ, 240^\circ, 330^\circ )", "Imaginary parts: ( \sin(60^\circ), \sin(150^\circ), \sin(240^\circ), \sin(330^\circ) )\nMaximum at ( \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 )", "Compare with earlier maximum:\n- From Case 1: ( \sin(97.5^\circ) \approx \sin(90^\circ + 7.5^\circ) = \cos(7.5^\circ) \approx 0.991 )", "Wait — correction: ( \sin(97.5^\circ) = \sin(180^\circ - 82.5^\circ) = \sin(82.5^\circ) \approx 0.991 ), which is actually larger than ( \sin(60^\circ) )", "So, although ( \sin(97.5^\circ) = \sin(82.5^\circ) ), it's approximately ( 0.991 ), significantly larger than ( 0.866 )", "Thus, the maximum imaginary part is ( \sin(97.5^\circ) )", "But wait — is this valid? Let's double-check whether ( 97.5^\circ ) root has positive imaginary part and is valid.", "Indeed, ( z = e^{i97.5^\circ} ) lies in second quadrant, where sine is positive and near maximum.", "But ( \sin(97.5^\circ) = \sin(180^\circ - 82.5^\circ) = \sin(82.5^\circ) ), and ( \sin(97.5^\circ) \approx 0.9914 )", "But let's now consider all roots and their imaginary parts:", "From Case 1 ((4\pi i/3)):\nMax imaginary part is ( \sin(97.5^\circ) \approx 0.9914 )", "From Case 2 ((4\pi i/3 + 2k\pi)) — all angles yield smaller or negative imaginary parts.", "Hence, the maximum imaginary part is ( \sin(97.5^\circ) ), so ( \phi = 97.5^\circ )", "But wait — problem states: “the maximum imaginary part can be expressed as ( \sin \phi ), where ( 0^\circ < \phi < 90^\circ )”", "But ( \sin \phi = \sin(180^\circ - \ heta) ), and the sine function is symmetric.", "Thus, although ( \sin(97.5^\circ) = \sin(82.5^\circ) ), and ( 82.5^\circ < 90^\circ ), we are asked to express the maximum imaginary part as ( \sin \phi ), and ( \phi ) must satisfy ( 0^\circ < \phi < 90^\circ )", "But the imaginary part is a value, not an angle — so to write ( \sin(97.5^\circ) ), we can use the identity:", "[\n\sin(97.5^\circ) = \sin(180^\circ - 82.5^\circ) = \sin(82.5^\circ)\n]", "So we may let ( \phi = 82.5^\circ ), since ( \sin \phi = \sin(97.5^\circ) ), and ( 0^\circ < 82.5^\circ < 90^\circ )", "But is there a simpler or exact expression?", "Wait — recall:\nThe roots from Case 1 are at angles:", "[\n7.5^\circ, 97.5^\circ, 187.5^\circ, 277.5^\circ\n]", "Their imaginary parts:", "[\n\sin(7.5^\circ),\ \sin(97.5^\circ),\ \sin(187.5^\circ) = -\sin(7.5^\circ),\ \sin(277.5^\circ) = -\sin(82.5^\circ)\n]", "So maximum is ( \sin(97.5^\circ) )", "But ( \sin(97.5^\circ) = \sin(82.5^\circ) ), so both represent the same value.", "But the problem says “can be expressed as ( \sin \phi )” with ( 0^\circ < \phi < 90^\circ ), so any angle ( \phi ) such that ( \sin \phi = \ ext{max imaginary part} ), and we are to find ( \phi )", "So the maximum value is ( \sin(97.5^\circ) ), but since sine is symmetric, we can also write ( \sin(82.5^\circ) )", "But ( \phi ) must be the angle such that ( \sin \phi = \ ext{max imaginary part} ), and since imaginary parts are positive, ( \phi \in (0^\circ, 90^\circ) )", "Thus, the correct choice is:", "[\n\phi = 82.5^\circ\n]", "But wait — is ( \sin(97.5^\circ) = \sin \phi \Rightarrow \phi = 82.5^\circ ), since ( 180^\circ - 97.5^\circ = 82.5^\circ )", "Hence, ( \phi = 82.5^\circ )", "But let’s verify:", "Is there a root with imaginary part ( \sin \phi ), where ( \phi = 82.5^\circ )? Yes — ( \phi = 97.5^\circ ) gives ( \sin = \sin(82.5^\circ) ), but we want ( \phi ) such that the maximum imaginary part equals ( \sin \phi ), so ( \sin \phi = \sin(97.5^\circ) ), so ( \phi = 97.5^\circ ) is valid, but the constraint restricts ( \phi < 90^\circ ), so we must find an equivalent acute angle.", "But since ( \sin(97.5^\circ) = \sin(82.5^\circ) ), and ( 82.5^\circ < 90^\circ ), we can take:", "[\n\phi = 82.5^\circ\n]", "Alternatively, is ( \sin(97.5^\circ) ) equal to ( \sin(82.5^\circ) )? Yes — because:", "[\n\sin(97.5^\circ) = \sin(180^\circ - 82.5^\circ) = \sin(82.5^\circ)\n]", "So ( \phi = 82.5^\circ ) satisfies ( \sin \phi = \ ext{max imaginary part} ), and ( 0^\circ < 82.5^\circ < 90^\circ )", "But is there a better exact form?", "Wait — can we express ( \sin(97.5^\circ) ) cleanly?", "Note:\n( 97.5^\circ = 90^\circ + 7.5^\circ ), so ( \sin(97.5^\circ) = \cos(7.5^\circ) ), but that doesn’t help directly.", "But geometrically, among the roots, the maximum imaginary part is ( \sin(97.5^\circ) ), and since this equals ( \sin(82.5^\circ) ), and both are valid, but the angle in ( (0^\circ, 90^\circ) ) whose sine is the value is any such angle. The problem asks to find ( \phi ) such that the imaginary part equals ( \sin \phi ), so any such ( \phi ) is acceptable, but likely expects the acute angle representing the value.", "However, conventionally, we write the angle in ( (0^\circ, 90^\circ) ) with the sine equal to the max part. Since:", "[\n\sin(98^\circ) \approx \sin(82^\circ)\n]", "But exact value: ( \sin(97.5^\circ) = \sin\left(\frac{165^\circ}{2}\right) ), but better:", "We can use:", "[\n\sin(97.5^\circ) = \sin\left(90^\circ + 7.5^\circ\right) = \cos(7.5^\circ)\n]", "But ( \cos(7.5^\circ) ) is not a standard angle, so expressing ( \phi = 82.5^\circ ) is acceptable and exact.", "But wait — is ( \sin(97.5^\circ) = \sin(82.5^\circ) )? Yes.", "So ( \phi = 82.5^\circ ) satisfies the requirement.", "But is there a real degree angle less than 90° with ( \sin \phi = \sin(97.5"]