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$ rac{pi}{3}$

Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 2025

Definition

$ rac{pi}{3}$ is a mathematical constant that represents one-third of the value of the mathematical constant pi (π), which is approximately equal to 3.14159. This term is particularly important in the context of the unit circle, as it represents a specific angle measure and the corresponding coordinates on the unit circle.

$ rac{pi}{3}$ cheat sheet for homework

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5 Must Know Facts For Your Next Test

  1. $ rac{pi}{3}$ is an angle measure that is equal to 60 degrees or approximately 1.0472 radians.
  2. On the unit circle, the coordinates for the angle $ rac{pi}{3}$ are $ extbackslash left( rac{ extbackslash sqrt{3}}{2}, rac{1}{2} extbackslash right)$.
  3. The angle $ rac{pi}{3}$ is often used in the study of special triangles, such as the 30-60-90 triangle, which has side lengths in the ratio of 1:$ extbackslash sqrt{3}$:2.
  4. The sine and cosine values for the angle $ rac{pi}{3}$ are $ rac{1}{2}$ and $ rac{ extbackslash sqrt{3}}{2}$, respectively.
  5. The tangent value for the angle $ rac{pi}{3}$ is $ extbackslash sqrt{3}$.

Review Questions

  • Explain the significance of the angle $ rac{pi}{3}$ in the context of the unit circle.
    • The angle $ rac{pi}{3}$ is significant in the context of the unit circle because it represents a specific angle measure that is used to study the behavior of trigonometric functions. On the unit circle, the coordinates for the angle $ rac{pi}{3}$ are $ extbackslash left( rac{ extbackslash sqrt{3}}{2}, rac{1}{2} extbackslash right)$, which are important in understanding the relationships between the trigonometric functions and the unit circle. Additionally, the angle $ rac{pi}{3}$ is often used in the study of special triangles, such as the 30-60-90 triangle, which has important applications in various areas of mathematics and physics.
  • Describe the relationship between the angle $ rac{pi}{3}$ and the trigonometric functions.
    • The angle $ rac{pi}{3}$ has specific values for the trigonometric functions of sine, cosine, and tangent. The sine value for $ rac{pi}{3}$ is $ rac{1}{2}$, the cosine value is $ rac{ extbackslash sqrt{3}}{2}$, and the tangent value is $ extbackslash sqrt{3}$. These values are important in understanding the behavior of the trigonometric functions and their relationships with angles on the unit circle. Additionally, the angle $ rac{pi}{3}$ is often used in the study of special triangles, which further highlights the connections between the angle, the trigonometric functions, and their applications in mathematics and physics.
  • Analyze the significance of the angle $ rac{pi}{3}$ in the context of radian measure and its applications.
    • The angle $ rac{pi}{3}$ is significant in the context of radian measure because it represents one-third of the value of the mathematical constant pi (π), which is approximately equal to 3.14159. Radian measure is a way of measuring angles that is closely tied to the unit circle and the behavior of trigonometric functions. The angle $ rac{pi}{3}$, expressed in radians, is approximately equal to 1.0472 radians. This angle measure is important in understanding the relationships between angles, the unit circle, and the trigonometric functions, which have widespread applications in fields such as mathematics, physics, engineering, and beyond.
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