tan(8π/3) Trigonometric Functions Formula Overview

This article provides a comprehensive overview of the trigonometric function formulas related to tan(8π/3). It highlights the properties, definitions, and transformations associated with this specific angle, giving detailed insights into its implications in trigonometric calculations.

Understanding tan(8π/3)

To understand the value of tan(8π/
3), it is essential to express this angle in a more manageable form. The angle 8π/3 can be simplified by subtracting 2π, which is a full revolution. By doing this, we find: 8π/3 - 2π = 8π/3 - 6π/3 = 2π/3. Therefore, tan(8π/3) is equivalent to tan(2π/3). The angle of 2π/3 is located in the second quadrant where the tangent function is negative. Hence, we determine the value of tan(2π/3) to find the final result for tan(8π/3).

Properties of Tangent Function

The tangent function has several important properties that are useful for calculations involving angles. Firstly, the tangent of the complementary angle has the relation: tan(θ) = -tan(π - θ). For θ = 2π/
3, this can be applied to emphasize the sign change that occurs in different quadrants. Another notable property is the periodic nature of the tangent function: tan(θ + nπ) = tan(θ) for any integer n, which illustrates that the tangent function repeats every π radians. Therefore, understanding these properties is fundamental when working with various angles, including tan(8π/3).

Calculating Trigonometric Values for tan(2π/3)

To compute tan(2π/
3), we can use the unit circle or trigonometric identities. The coordinates at the angle 2π/3 are (-1/
2, √3/2). The tangent function, defined as the ratio of the sine to the cosine, yields: tan(2π/3) = sin(2π/3) / cos(2π/3) = (√3/2) / (-1/2) = -√3. Therefore, we conclude that tan(8π/3) = -√3. This clarity and simplification of angles demonstrate the effectiveness of employing fundamental trigonometric identities.

In summary, the angle tan(8π/3) can be simplified to tan(2π/
3), which is evaluated to -√3. Understanding the properties of tangents and their periodic nature aids in effectively navigating through trigonometric calculations. By breaking down complex angles and applying foundational concepts, we gain valuable insights into trigonometric functions.

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