Exploring the Value of Tan 3πx

Tan 2πx and Tan 3πx Explained:

Tan is a periodic function with a period of π, meaning that for any tan(x 2π) tan(x). In the context of the question, tan(2πx) tan(x). However, without a specific value for x, it is not possible to determine the exact value of the tangent.

For the case of tan(3πx), the situation is slightly different. When we add 3π to the angle, we can break it down into simpler parts to understand its behavior.

Given that 2πx will lie in the first quadrant (since the tangent function is periodic, and 2π is a full cycle), the angle 3πx can be decomposed into 2π πx. Here, 2π represents a full cycle, meaning that it does not affect the value of the tangent. Thus, the remaining angle is π x.

The angle π x lies in the third quadrant because:

First Quadrant: 0 to π/2 (0° to 90°) Second Quadrant: π/2 to π (90° to 180°) Third Quadrant: π to 3π/2 (180° to 270°) Fourth Quadrant: 3π/2 to 2π (270° to 360°)

Since 2π πx places the angle in the first quadrant and π x places it in the third quadrant, the value of the tangent in the third quadrant is positive. Thus, we can conclude that:

tan(3πx) tan(π x) -tan(x)

But since we are in the third quadrant, the result simplifies to:

tan(3πx) tan(x)

Key Points:

Periodicity of the tangent function Effect of adding 2π to an angle Behavior of tangent in different quadrants

Conclusion:

Thus, the value of tan(3πx) is tan(x). The final answer is positive because the tangent function is positive in the third quadrant.

Further Reading:

For a deeper understanding of trigonometric functions and periodicity, consider exploring related topics:

Trigonometry Overview Visual Guide to Trigonometry Understanding Trigonometric Periodicity