From 77e7956b788189ab1dd44812f0396a31031470f3 Mon Sep 17 00:00:00 2001
From: z3rOR0ne <z3rOR0ne@protonmail.com>
Date: Sat, 26 Apr 2025 07:02:21 -0700
Subject: [PATCH] :memo: Made note of unit circle values

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 mathematics/unit_circle_trig_tables.md | 58 ++++++++++++++++++++++++++
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 create mode 100644 mathematics/unit_circle_trig_tables.md

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+# Unit Circle Trigonometric Values
+
+## Key Angles in Degrees and Radians:
+
+- $0^\circ = 0$ radians
+- $30^\circ = \frac{\pi}{6}$ radians
+- $45^\circ = \frac{\pi}{4}$ radians
+- $60^\circ = \frac{\pi}{3}$ radians
+- $90^\circ = \frac{\pi}{2}$ radians
+- $120^\circ = \frac{2\pi}{3}$ radians
+- $135^\circ = \frac{3\pi}{4}$ radians
+- $150^\circ = \frac{5\pi}{6}$ radians
+- $180^\circ = \pi$ radians
+- $210^\circ = \frac{7\pi}{6}$ radians
+- $225^\circ = \frac{5\pi}{4}$ radians
+- $240^\circ = \frac{4\pi}{3}$ radians
+- $270^\circ = \frac{3\pi}{2}$ radians
+- $300^\circ = \frac{5\pi}{3}$ radians
+- $315^\circ = \frac{7\pi}{4}$ radians
+- $330^\circ = \frac{11\pi}{6}$ radians
+- $360^\circ = 2\pi$ radians
+
+## Trigonometric Values at Key Angles:
+
+| Angle (Degrees) | Angle (Radians)   | $ \cos(\theta) $      | $ \sin(\theta) $      | $ \tan(\theta) $      |
+| --------------- | ----------------- | --------------------- | --------------------- | --------------------- |
+| $0^\circ$       | $0$               | $1$                   | $0$                   | $0$                   |
+| $30^\circ$      | $\frac{\pi}{6}$   | $\frac{\sqrt{3}}{2}$  | $\frac{1}{2}$         | $\frac{1}{\sqrt{3}}$  |
+| $45^\circ$      | $\frac{\pi}{4}$   | $\frac{\sqrt{2}}{2}$  | $\frac{\sqrt{2}}{2}$  | $1$                   |
+| $60^\circ$      | $\frac{\pi}{3}$   | $\frac{1}{2}$         | $\frac{\sqrt{3}}{2}$  | $\sqrt{3}$            |
+| $90^\circ$      | $\frac{\pi}{2}$   | $0$                   | $1$                   | $\text{undefined}$    |
+| $120^\circ$     | $\frac{2\pi}{3}$  | $-\frac{1}{2}$        | $\frac{\sqrt{3}}{2}$  | $-\sqrt{3}$           |
+| $135^\circ$     | $\frac{3\pi}{4}$  | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$  | $-1$                  |
+| $150^\circ$     | $\frac{5\pi}{6}$  | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$         | $-\frac{1}{\sqrt{3}}$ |
+| $180^\circ$     | $\pi$             | $-1$                  | $0$                   | $0$                   |
+| $210^\circ$     | $\frac{7\pi}{6}$  | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$        | $\frac{1}{\sqrt{3}}$  |
+| $225^\circ$     | $\frac{5\pi}{4}$  | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $1$                   |
+| $240^\circ$     | $\frac{4\pi}{3}$  | $-\frac{1}{2}$        | $-\frac{\sqrt{3}}{2}$ | $\sqrt{3}$            |
+| $270^\circ$     | $\frac{3\pi}{2}$  | $0$                   | $-1$                  | $\text{undefined}$    |
+| $300^\circ$     | $\frac{5\pi}{3}$  | $\frac{1}{2}$         | $-\frac{\sqrt{3}}{2}$ | $-\sqrt{3}$           |
+| $315^\circ$     | $\frac{7\pi}{4}$  | $\frac{\sqrt{2}}{2}$  | $-\frac{\sqrt{2}}{2}$ | $-1$                  |
+| $330^\circ$     | $\frac{11\pi}{6}$ | $\frac{\sqrt{3}}{2}$  | $-\frac{1}{2}$        | $\frac{1}{\sqrt{3}}$  |
+| $360^\circ$     | $2\pi$            | $1$                   | $0$                   | $0$                   |
+
+## Patterns to Notice:
+
+1. **Sine and Cosine Symmetry:**
+   - The values of sine and cosine are symmetric in different quadrants.
+   - For example, sine values are positive in the first and second quadrants,
+     negative in the third and fourth quadrants.
+   - Cosine values are positive in the first and fourth quadrants, negative in
+     the second and third quadrants.
+
+2. **Tangent Behavior:**
+   - Tangent values are undefined when cosine equals zero (at $90^\circ$ and
+     $270^\circ$).
+   - Tangent has a period of $180^\circ$, so the values for angles like
+     $45^\circ$ and $225^\circ$ are the same.