The exact value of sin(pi/3) is √3/2. This fundamental trigonometric value is frequently encountered in mathematics and various scientific fields.
Understanding pi/3 in Degrees
To determine the value of sin(pi/3), it's helpful to convert the angle from radians to degrees. The conversion factor is 180°/π radians.
- π/3 radians = (π/3) * (180°/π) = 60°
Therefore, finding the value of sin(pi/3) is equivalent to finding the value of sin(60°). Both expressions represent the same angle and thus have the same sine value.
The Exact Value of sin(60°)
The sine of 60 degrees (or pi/3 radians) is a special trigonometric value derived from the properties of a 30-60-90 right triangle or the unit circle.
| Angle (Radians) | Angle (Degrees) | Exact Sine Value | Approximate Decimal Value |
|---|---|---|---|
| pi/3 | 60° | √3/2 | 0.8660254... |
This value, √3/2, is considered the exact answer, while 0.8660254 is its decimal approximation, often rounded to fewer decimal places depending on the required precision.
For more information on special angles in trigonometry, you can refer to resources on the unit circle or special right triangles.
Applications of sin(pi/3)
The value of sin(pi/3) is crucial in many areas, including:
- Geometry: Calculating side lengths and angles in triangles.
- Physics: Analyzing wave motion, projectile trajectories, and forces in vector components.
- Engineering: Designing structures, circuits, and mechanical systems where angular relationships are important.
- Calculus: Evaluating integrals and derivatives involving trigonometric functions.
Understanding these foundational trigonometric values provides a strong basis for solving complex problems across various disciplines.
