Find the sum of all angles \(z \in [0^\circ, 360^\circ]\) that satisfy \(\sin z + \cos z = 1\).

["Title: Find All Angles (z) Between (0^\circ) and (360^\circ) That Satisfy (\sin z + \cos z = 1)", "---", "Introduction\nFinding angles ( z ) within the range ( [0^\circ, 360^\circ] ) such that ( \sin z + \cos z = 1 ) is a classic trigonometric challenge with broad applications in physics, engineering, and mathematics. This article walks you through solving the equation step-by-step, explaining transformations and key identities—helping you confidently identify all valid solutions.", "---", "### Step 1: Rewrite the Equation Using a Single Trigonometric Function", "The equation\n[\n\sin z + \cos z = 1\n]\ncan be rewritten in a more manageable form using a trigonometric identity. The expression ( \sin z + \cos z ) is equivalent to:\n[\n\sin z + \cos z = \sqrt{2} \sin\left(z + 45^\circ\right)\n]\nThis transformation uses the identity:\n[\n\sin z + \cos z = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin z + \frac{1}{\sqrt{2}} \cos z \right) = \sqrt{2} \sin\left(z + 45^\circ\right)\n]\nSo the equation becomes:\n[\n\sqrt{2} \sin(z + 45^\circ) = 1\n]\nDivide both sides by ( \sqrt{2} ):\n[\n\sin(z + 45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\n]", "---", "### Step 2: Solve for ( z + 45^\circ )", "We now solve:\n[\n\sin \ heta = \frac{\sqrt{2}}{2}\n]\nwhere ( \ heta = z + 45^\circ ).", "The general solutions for ( \sin \ heta = \frac{\sqrt{2}}{2} ) are:\n[\n\ heta = 45^\circ + 360^\circ k \quad \ ext{or} \quad \ heta = 135^\circ + 360^\circ k, \quad \ ext{for integer } k\n]", "Substitute back ( \ heta = z + 45^\circ ):\n1. ( z + 45^\circ = 45^\circ + 360^\circ k \Rightarrow z = 360^\circ k )\n2. ( z + 45^\circ = 135^\circ + 360^\circ k \Rightarrow z = 90^\circ + 360^\circ k )", "---", "### Step 3: Restrict Solutions to ( z \in [0^\circ, 360^\circ] )", "Now determine which of these satisfy ( z \in [0^\circ, 360^\circ] ):", "- From ( z = 360^\circ k ):\n - When ( k = 0 ): ( z = 0^\circ ) ✅\n - When ( k = 1 ): ( z = 360^\circ ) ✅ (included in closed interval)", "- From ( z = 90^\circ + 360^\circ k ):\n - When ( k = 0 ): ( z = 90^\circ ) ✅\n - When ( k = 1 ): ( z = 450^\circ > 360^\circ ) ❌", "So the candidate solutions are ( z = 0^\circ, 90^\circ, 360^\circ ).", "However, since ( 0^\circ ) and ( 360^\circ ) represent the same angle on the unit circle, in most mathematical contexts they are considered equivalent. But since the interval is closed at both ends, both endpoints are valid and distinct solutions in this context.", "Thus, the distinct solutions in ( [0^\circ, 360^\circ] ) are:\n[\nz = 0^\circ, \quad z = 90^\circ, \quad z = 360^\circ\n]", "---", "### Step 4: Verify Each Solution in the Original Equation", "Plug each value into ( \sin z + \cos z = 1 ):", "- For ( z = 0^\circ ):\n [\n \sin 0^\circ + \cos 0^\circ = 0 + 1 = 1 \quad \ ext{✓}\n ]", "- For ( z = 90^\circ ):\n [\n \sin 90^\circ + \cos 90^\circ = 1 + 0 = 1 \quad \ ext{✓}\n ]", "- For ( z = 360^\circ ):\n [\n \sin 360^\circ + \cos 360^\circ = 0 + 1 = 1 \quad \ ext{✓}\n ]", "All three values satisfy the equation.", "---", "### Conclusion", "The angles ( z ) in the interval ( [0^\circ, 360^\circ] ) that satisfy ( \sin z + \cos z = 1 ) are:\n[\n\boxed{0^\circ,\ 90^\circ,\ 360^\circ}\n]", "---", "### Key Takeaways\n- Use identities like ( \sin z + \cos z = \sqrt{2} \sin(z + 45^\circ) ) to simplify trigonometric equations.\n- Solve transformed equations within one period, then restrict to the desired interval.\n- Verify solutions to avoid extraneous results from algebraic manipulation.\n- Understanding angle periodicity explains why ( 0^\circ ) and ( 360^\circ ) are distinct yet equivalent solutions.", "This method applies broadly to trigonometric problems in exams, engineering calculations, and real-world modeling.", "---", "Tags: #Trigonometry #SolveEquations #MathTips #sinCos #AnglesInDegrees #0To360 #TrigonometricSolutions"]