π/6

Definition

π/6 is a significant angle in trigonometry, corresponding to 30 degrees. It is one of the key angles found in the unit circle, where it plays a crucial role in defining the sine and cosine values of common angles. Understanding π/6 helps in grasping how the circular functions behave and relate to the geometry of the unit circle.

5 Must Know Facts For Your Next Test

1. The coordinates for the point on the unit circle at π/6 are (√3/2, 1/2), where √3/2 is the cosine value and 1/2 is the sine value.
2. π/6 radians corresponds to a 30-degree angle, making it easy to remember when working with standard angles.
3. The tangent of π/6 can be calculated as sin(π/6) / cos(π/6), resulting in a value of 1/√3 or √3/3.
4. In the context of special triangles, π/6 relates to the 30-60-90 triangle, where the sides are in the ratio 1 : √3 : 2.
5. The periodicity of trigonometric functions means that π/6 also appears in other quadrants, influencing their values based on their respective angles.

Review Questions

• How does understanding π/6 enhance your ability to work with other angles on the unit circle?
  • Knowing π/6 helps you understand how to derive values for other angles on the unit circle because it is a reference angle. For instance, angles like π/3 and 5π/6 can be analyzed by relating them back to π/6, allowing you to use symmetry and properties of sine and cosine. This foundational knowledge gives insight into how trigonometric values repeat based on their positions around the unit circle.
• In what ways do sine and cosine values at π/6 compare to those at other key angles, such as π/4 and π/3?
  • At π/6, sine and cosine values differ significantly from those at other angles. The sine value at π/6 is 1/2, while at π/4 it is √2/2 and at π/3 it is √3/2. Similarly, cosine shows variation, with values of √3/2 at π/6, √2/2 at π/4, and 1/2 at π/3. These comparisons demonstrate how each angle affects trigonometric ratios differently based on their respective positions within the unit circle.
• Evaluate how knowing the properties of the angle π/6 aids in solving more complex trigonometric equations.
  • Understanding properties of π/6 allows you to simplify complex equations by breaking them down into manageable parts. For example, when dealing with equations involving multiple angles or identities, you can replace instances of sin(θ) or cos(θ) with their values from π/6. Additionally, knowing that these functions are periodic means you can find solutions across different quadrants by adding or subtracting multiples of π, making it easier to handle a wide range of problems effectively.

Related terms

Unit Circle: A circle with a radius of one, centered at the origin of a coordinate plane, used to define trigonometric functions.

Sine: A trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.

Cosine: A trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle.