Exploring the Tangent of Small Angles: 1°, 2°, ..., 9°

Understanding the tangent function for small angles like 1°, 2°, ..., 9° is a fascinating aspect of trigonometry. In this article, we will delve into the properties of the tangent function for these specific angles and explore the mathematical techniques to understand and compute the sum of these tangents. This discussion will not only provide a clear explanation but also serve as a useful reference for students and mathematicians interested in trigonometry.

The Sum of Small Angles

Before we proceed, let us first establish a fundamental approach to summing these angles. The sum of the first n natural numbers can be calculated using the formula:

S n(n 1)/2

This formula helps in determining the sum of the angles 1°, 2°, ..., 9°.

Calculating the Sum of the Angles

Given that n 9, we can apply the sum formula:

S 9(9 1)/2

Perform the arithmetic operations:

S 9(10)/2 45°

Thus, the sum of the angles 1°, 2°, ..., 9° equals 45°. This is a crucial step in understanding the tangent of these small angles.

Tan of 45 Degrees

We now know that the sum of the angles is 45°. It is well-known that the tangent of 45° is equal to 1:

tan(45°) 1

Given the properties of the tangent function, particularly for standard angles, we can infer that the individual tangents of the smaller angles (1°, 2°, ..., 9°) contribute to this sum in a way that their aggregate effect equals the tangent of 45°.

Conclusion

In conclusion, we have determined that:

tan(1°) tan(2°) ... tan(9°) 1

This result is a fascinating application of trigonometric principles and the sum of natural numbers. For those interested in further exploration, this property can be used as a basis to understand and compute tangents for other sets of small angles.

Related Topics

1. Tangent of Small Angles

2. tan 1 Degree

3. Sum of Natural Numbers