Understanding the Tangent of 250 Degrees: A Comprehensive Guide

This guide delves into the concept of the tangent of 250 degrees, discussing the periodicity of the tangent function, the use of reference angles, and the exact and approximate values of the tangent of this angle. The information provided is designed to be easy to understand and follow, ensuring that you gain a thorough understanding of the topic.

Introduction to the Tangent Function and Periodicity

The tangent function, denoted as tan(θ), is a fundamental trigonometric function that represents the ratio of the sine to the cosine of an angle. It has a inherent periodicity, meaning tan(θ) tan(θ 180°).

Using Periodicity to Simplify the Problem

To find the value of tan(250°), we can use the periodicity of the tangent function. The periodicity allows us to reduce the angle to a more manageable range of 0° to 180°. The transformation can be done as follows:

Calculate the reference angle: 250° - 180° 70°. This angle is the smallest positive angle that gives the same tangent value as 250°. Identify the quadrant of the angle 250°. Since 250° is in the third quadrant, the tangent value will be positive. The tangent function is positive in the first and third quadrants. Thus, tan(250°) tan(70°).

Using a calculator or a trigonometric table, we can find the numerical value for tan(70°) ≈ 2.747. Therefore, tan(250°) ≈ 2.747.

Exact Value Calculation

The exact value of tan(250°) can be derived using complex numbers and Euler's formula. The exact value is given by the expression:

tan(250°) (e^8/9πe^-iπ/9)/e25/18π-e^11/18π

This is a complex expression, and the real part of this value is positive. For simplicity, we can use software like WolframAlpha to find the numerical approximation. WolframAlpha can solve the cubic equation derived from the tangent triple angle formula:

tan(3θ) tan(θ)/3tan^2(θ) - 3/tan(θ) - 1/3

By solving this equation for θ 250°, we can find the approximate value:

tan(250°) ≈ 2.7475

Use of Reference Angles and Quadrant Rules

Reference angles and quadrant rules are also useful for understanding the tangent of 250°. Here are the rules for reference angles and quadrants:

First Quadrant (0° to 90°): 0° ≤ θ 90° Second Quadrant (90° to 180°): 90° ≤ θ 180° Third Quadrant (180° to 270°): 180° ≤ θ 270° Fourth Quadrant (270° to 360°): 270° ≤ θ 360°

For 250°, which is in the third quadrant, the tangent value is positive. We can further simplify this by recognizing that 250° 180° 70°. Thus, tan(250°) tan(70°), which is in the first quadrant.

Conclusion

In conclusion, understanding the tangent of 250 degrees requires a thorough understanding of periodicity, reference angles, and quadrant rules. By using these tools, we can find both the exact and approximate values of the tangent of this angle. This guide has provided the necessary steps and explanations to achieve this understanding.