The exact sine of pi/6 (π/6) is 1/2, and the exact cosine of pi/6 (π/6) is √3/2. These values are fundamental in trigonometry and are derived from the unit circle, representing the coordinates of the angle.
Exact Trigonometric Values for Pi/6
The angle pi/6 radians is equivalent to 30 degrees. This is one of the most frequently encountered angles in trigonometry due to its precise and easy-to-remember exact values for sine, cosine, and tangent.
Here's a quick reference for π/6:
| Angle (Radians) | Angle (Degrees) | Sine (sin) | Cosine (cos) |
|---|---|---|---|
| π/6 | 30° | 1/2 | √3/2 |
Understanding Pi/6 on the Unit Circle
The unit circle is a circle with a radius of one unit, centered at the origin (0,0) of a Cartesian coordinate system. It's a powerful tool for understanding trigonometric functions. For any point (x, y) on the unit circle corresponding to an angle $\theta$ (measured counter-clockwise from the positive x-axis):
- The x-coordinate of the point is equal to the cosine of the angle,
cos(θ). - The y-coordinate of the point is equal to the sine of the angle,
sin(θ).
For π/6 (or 30°), the point on the unit circle is located at approximately (0.866, 0.5). In exact terms, this point is (√3/2, 1/2).
- This means
cos(π/6) = √3/2 - And
sin(π/6) = 1/2
To learn more about the unit circle and its properties, you can explore resources like Khan Academy's Unit Circle Introduction or Math is Fun's Unit Circle Guide.
Common Trigonometric Ratios
Knowing the exact trigonometric values for common angles like π/6, π/4, and π/3 is crucial for solving various mathematical and real-world problems. Here's a table summarizing these key ratios, including their tangent values:
| Angle (Radians) | Angle (Degrees) | Sine (sin) | Cosine (cos) | Tangent (tan) |
|---|---|---|---|---|
| π/6 | 30° | 1/2 | √3/2 | √3/3 |
| π/4 | 45° | √2/2 | √2/2 | 1 |
| π/3 | 60° | √3/2 | 1/2 | √3 |
Applications and Importance
These exact trigonometric values are not just theoretical; they have practical applications across various fields:
- Physics: Used in calculations involving forces, waves, and motion.
- Engineering: Essential for design, stress analysis, and signal processing.
- Computer Graphics: Fundamental for transformations and rendering.
- Navigation: Critical for calculating positions and directions.
By understanding and memorizing these exact values, you gain a strong foundation for advanced mathematical concepts and their real-world applications.
