Unit Circle Values
Verify unit circle values for standard angles using trigonometric functions and the pi constant. Reference table of sin, cos, and tan for common radian values.
Trigonometry
Detailed Explanation
Unit Circle Values
The unit circle is a circle with radius 1 centered at the origin. Every point on the circle can be described as (cos(theta), sin(theta)).
Standard Angles
You can verify every value using the evaluator:
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0 | sin(0) = 0 |
cos(0) = 1 |
tan(0) = 0 |
| 30 | pi/6 | sin(pi/6) = 0.5 |
cos(pi/6) = 0.866 |
tan(pi/6) = 0.577 |
| 45 | pi/4 | sin(pi/4) = 0.707 |
cos(pi/4) = 0.707 |
tan(pi/4) = 1 |
| 60 | pi/3 | sin(pi/3) = 0.866 |
cos(pi/3) = 0.5 |
tan(pi/3) = 1.732 |
| 90 | pi/2 | sin(pi/2) = 1 |
cos(pi/2) = 0 |
tan(pi/2) = undefined |
| 180 | pi | sin(pi) = 0 |
cos(pi) = -1 |
tan(pi) = 0 |
| 270 | 3*pi/2 | sin(3*pi/2) = -1 |
cos(3*pi/2) = 0 |
undefined |
| 360 | 2*pi | sin(2*pi) = 0 |
cos(2*pi) = 1 |
tan(2*pi) = 0 |
Key Identities to Verify
Pythagorean Identity:
sin(pi/4)^2 + cos(pi/4)^2 = 1
Double Angle:
2*sin(pi/6)*cos(pi/6) = sin(pi/3)
Both sides equal approximately 0.866.
Complementary Angles:
sin(pi/6) = cos(pi/3) = 0.5
sin(pi/3) = cos(pi/6) = 0.866
Exact vs Decimal Values
The evaluator returns decimal approximations. The exact values involve square roots:
- sin(pi/4) = sqrt(2)/2 (verify:
sqrt(2)/2= 0.70710...) - sin(pi/3) = sqrt(3)/2 (verify:
sqrt(3)/2= 0.86602...) - sin(pi/6) = 1/2 (verify: exactly 0.5)
Use Case
A trigonometry student verifying unit circle values from a textbook by computing each sin, cos, and tan value directly in the evaluator.