3.4 KiB
3.4 KiB
Unit Circle Trigonometric Values
Key Angles in Degrees and Radians:
0^\circ = 0radians30^\circ = \frac{\pi}{6}radians45^\circ = \frac{\pi}{4}radians60^\circ = \frac{\pi}{3}radians90^\circ = \frac{\pi}{2}radians120^\circ = \frac{2\pi}{3}radians135^\circ = \frac{3\pi}{4}radians150^\circ = \frac{5\pi}{6}radians180^\circ = \piradians210^\circ = \frac{7\pi}{6}radians225^\circ = \frac{5\pi}{4}radians240^\circ = \frac{4\pi}{3}radians270^\circ = \frac{3\pi}{2}radians300^\circ = \frac{5\pi}{3}radians315^\circ = \frac{7\pi}{4}radians330^\circ = \frac{11\pi}{6}radians360^\circ = 2\piradians
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:
-
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.
-
Tangent Behavior:
- Tangent values are undefined when cosine equals zero (at
90^\circand270^\circ). - Tangent has a period of
180^\circ, so the values for angles like45^\circand225^\circare the same.
- Tangent values are undefined when cosine equals zero (at