Trigonometry Angles- From 0° to 360° Guide
What This Guide Covers
Every trigonometry problem eventually comes back to angles. 0°, 30°, 45°, 60°, 90°, and everything in between. If you're mixing up sine and cosine, or can't remember which quadrant gives positive or negative values, this is your reference.
Skip the theory lectures. Here's what actually matters.
The Unit Circle: Your Foundation
The unit circle is a circle with radius 1, centered at the origin. Every point on it follows the rules:
- x-coordinate = cosine of the angle
- y-coordinate = sine of the angle
Memorize this and half your trig problems solve themselves.
The Four Quadrants
Your 360° breaks into four sections. Each one has its own sign rules.
Quadrant I (0° to 90°)
Everything positive. Sine, cosine, tangent—all positive. This is the easy quadrant.
Quadrant II (90° to 180°)
Sine positive. Cosine and tangent negative. Only sine survives here.
Quadrant III (180° to 270°)
Tangent positive. Sine and cosine negative. Only tangent is happy here.
Quadrant IV (270° to 360°)
Cosine positive. Sine and tangent negative. Only cosine works here.
All Six Trig Functions
Most students know sine and cosine. They forget the other four.
- Sine (sin) = opposite / hypotenuse
- Cosine (cos) = adjacent / hypotenuse
- Tangent (tan) = opposite / adjacent = sin / cos
- Cosecant (csc) = hypotenuse / opposite = 1 / sin
- Secant (sec) = hypotenuse / adjacent = 1 / cos
- Cotangent (cot) = adjacent / opposite = 1 / tan = cos / sin
Remember: csc, sec, and cot are just reciprocals. If you know sin, cos, and tan, you know everything.
Special Angles and Their Values
These angles appear constantly. Know them cold.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
For 180°, 270°, and 360°, flip the signs based on the quadrant rules above.
How to Find Trig Values for Any Angle
Step 1: Identify which quadrant your angle lands in.
Step 2: Find the reference angle—the acute angle between your angle and the x-axis.
Step 3: Get the absolute value of the trig function from the special angles table.
Step 4: Apply the sign rules for your quadrant.
Example: Find sin 150°.
- 150° is in Quadrant II
- Reference angle = 180° - 150° = 30°
- sin 30° = ½
- Quadrant II: sine is positive
- Answer: sin 150° = ½
Reference Angles: The Shortcut
You only need to memorize values for 0° to 90°. Everything else derives from that.
- Quadrant I: reference angle = the angle itself
- Quadrant II: reference angle = 180° - angle
- Quadrant III: reference angle = angle - 180°
- Quadrant IV: reference angle = 360° - angle
That's it. One table of values covers all 360 degrees.
Common Mistakes
Confusing the sign rules. Draw a quick sketch. If you can't visualize the quadrants, you'll keep getting negative values wrong.
Forgetting tangent at 90° and 270°. Tangent = sin/cos. When cosine is zero, you're dividing by zero. Tangent doesn't exist there.
Using degrees when your calculator is in radians. Check your mode. Most algebra/trig problems use degrees. Calculus uses radians.
Memorizing without understanding. The unit circle isn't magic. It's just geometry. x = cos, y = sin on a circle with radius 1. That's all it means.
Quick Reference: Full Circle
| Angle | sin | cos | tan | Quadrant |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | I |
| 90° | 1 | 0 | undefined | I |
| 180° | 0 | -1 | 0 | II |
| 270° | -1 | 0 | undefined | III |
| 360° | 0 | 1 | 0 | IV |
Bookmark this. Practice the reference angle method until it's automatic. The rest follows.