Now find all $ z \in [0^\circ, 360^\circ] $:

["How to Find All $ z \in [0^\circ, 360^\circ] $: A Complete Guide", "When solving trigonometric equations involving angles, one common task is to determine all values of $ z $ within the interval $[0^\circ, 360^\circ]$ that satisfy a given condition—often expressed as $ z \in [0^\circ, 360^\circ] $. This guide explains how to systematically find all such angles, whether solving basic equations or tackling more complex trigonometric relationships.", "---", "### Understanding the Interval $[0^\circ, 360^\circ]$", "The interval $[0^\circ, 360^\circ]$ covers a full rotation in degrees, representing all possible angles in standardized position:", "- Starts at $ 0^\circ $ (positive x-axis direction),\n- Increasing cyclically to $ 360^\circ $, which coincides with $ 0^\circ $.", "This range is essential for solving trigonometric equations because it includes all possible directed angles used in sine, cosine, tangent, and their inverse functions.", "---", "### Basic Approach: Solving Trigonometric Equations for $ z $", "To find all $ z \in [0^\circ, 360^\circ] $ satisfying an equation such as:", "$$\n\sin z = x \quad \ ext{or} \quad \cos z = x \quad \ ext{or} \quad \ an z = x\n$$", "follow these steps:", "#### 1. Solve the equation algebraically\nUse identities and known values to find reference angles for $ z $. For example:", "- $ \sin z = x $ → Solutions come from reverse sine with angles in Quadrants I and II.\n- $ \cos z = x $ → Solutions from reverse cosine and symmetry.\n- $ \ an z = x $ → Use arctangent and consider periodicity of tangent.", "#### 2. Use the periodicity of trigonometric functions", "All basic trigonometric functions are periodic:", "- $ \sin z $ and $ \cos z $: period $ 360^\circ $\n- $ \ an z $: period $ 180^\circ $", "This means solutions repeat every full cycle, so finding solutions modulo $ 360^\circ $ covers all required $ z $.", "#### 3. Account for all quadrants", "For equations like $ \sin z = a $, there are generally two solutions in $[0^\circ, 360^\circ]$, except when $ |a| = 1 $, which yields one solution. Similarly, cosine equation $ \cos z = a $ also has two angles unless $ |a| = \pm 1 $.", "For tangent equations, since tangent is exponential ($ \ an z = \ an(z + n\cdot180^\circ) $), add $ 180^\circ $ multiples to the principal solution.", "---", "### Example: Find all $ z \in [0^\circ, 360^\circ] $ such that $ \cos z = -\frac{1}{2} $", "1. First, find the reference angle:\n $ \cos^{-1} \left( \frac{1}{2} \right) = 60^\circ $.\n So, $ z = 60^\circ $ and $ z = 360^\circ - 60^\circ = 300^\circ $ satisfy $ \cos z = -\frac{1}{2} $.", "2. Verify:\n $ \cos 60^\circ = \frac{1}{2} $, $ \cos 300^\circ = \frac{1}{2} $ → signs correct for negative.", "Solutions:\n$$\nz = 60^\circ \quad \ ext{and} \quad z = 300^\circ\n$$", "---", "### Advanced Tips", "- Graphical approach: Plot $ y = \sin z $, $ y = \cos z $, or $ y = \ an z $ and find intersection points within $[0^\circ, 360^\circ]$.\n- Use inverse trigonometric functions: Calculator inverse sine or arccos give principal values; add $ 360^\circ $ for all copies if needed.\n- ** remembers symmetry: Cosine is even, sine odd; tangent odd—use symmetry to reduce work.", "---", "### Summary", "Finding all $ z \in [0^\circ, 360^\circ] $ satisfying trigonometric equations:", "- Understand the cyclic interval fully\n- Solve the equation analytically using known identities and reference angles\n- Account for periodicity and quadrant behavior\n- Add all valid solutions modulo $ 360^\circ $", "By mastering these strategies, you can efficiently determine every solution $ z $ in degrees within a full rotation—critical for applications in oscillatory motion, wave physics, engineering, and computer graphics.", "---", "Keywords: find $ z \in [0^\circ, 360^\circ] $, solve trigonometric equations, all solutions $ z $, sine cosine tangent angles, trigonometry tutorial, arithmetic angles, periodic functions.", "---", "Need help solving a specific equation? Use inverse trig functions with quadrant analysis and periodicity correction to find all possible $ z $ in degrees."]