Understanding Tan 720° and Tan 765°: A Comprehensive Exploration

Understanding the value of trigonometric functions, specifically tangent, is crucial for various applications in mathematics and engineering. In this article, we will explore the values of tan 720° and 765° using the properties of periodicity and identity formulas. Let's break down the methods used to determine these values, making our exploration both educational and accessible.

The Value of Tan 720°

The tangent function, tan θ, is a periodic function with a period of 180°. This means that the values of tan θ repeat every 180°. However, the double angle identity can also be applied to help us find the value oftan 720°. Let's walk through the steps:

Double Angle Identity for Tangent:

tan 2x 2 / (cot x - tan x)

However, in the case of tan 720°, we can simplify the process by recognizing that 720° is a multiple of 360°, the period of the tangent function:

Since 720° 2 * 360°, and the tangent function is periodic with a period of 180°, we can further reduce this to:

tan 720° tan (360° * 2 - 360°) tan 360°

Since tan 360° 0, it follows that:

tan 720° 0

The Value of Tan 765°

Similarly, we can explore the value of tan 765°. Again, using the periodicity of the tangent function, we can reduce the angle to a simpler form:

Since the tangent function is periodic with a period of 180°, we can express 765° in terms of a simpler angle:

tan 765° tan (765° - 4 * 180°) tan 45°

And since tan 45° 1, we have:

tan 765° 1

This method demonstrates how understanding the periodic nature of trigonometric functions can simplify the process of finding specific values of tangent.

The Period of the Graph y tan x

The graph of the tangent function, y tan x, has a period of 180°. This means that the function repeats its values every 180°. Knowing this, we can use the periodicity to simplify the process of finding the values of tan 720° and 765°.

For tan 720°, we can directly recognize that:

tan 720° tan (4 * 180°) tan 0° 0

For tan 765°, we reduce the angle using the periodicity:

tan 765° tan (45° 4 * 180°) tan 45° 1

Conclusion

Understanding the periodic nature of the tangent function, as well as using basic trigonometric identities, can simplify the process of finding specific values. This article has provided a detailed exploration of tan 720° and 765°, demonstrating the importance of recognizing periodicity and using identities to simplify calculations.

Related Keywords:

Trigonometric functions, Periodicity, Tangent, Double Angle Identity