Geometric Proof of the Identity: sin2x 2 sinx cosx

In this article, we will explore a geometric proof of the trigonometric identity sin2x 2 sinx cosx. This proof relies on the properties of right triangles and the basic definitions of sine and cosine, while avoiding the use of Euler's formula, compound angle formulas, or Cartesian coordinates. Let's dive into the detailed geometric approach.

Proof Using Right Triangles

Consider a right triangle ΔABC with the following properties:

Angle A x Angle B 90° Angle C 90° - x

Let the length of the side opposite angle A (side a) be denoted as a. The length of the adjacent side (side b) can be denoted as b, and the hypotenuse c.

Using the Definitions of Sine and Cosine

By definition, we have:

sin x a/c cos x b/c

To find sin2x and cos2x, we will double the angle x to consider the angle 2x.

Double the Angle

We can visualize 2x by constructing another triangle with angle A 2x. Let's use the properties of the original right triangle to derive the expression for sin2x.

Using the Angle Addition Theorem

Although the angle addition theorem can be used to express sin2x as:

sin2x 2 sin x cos x

We will proceed geometrically without explicitly using this formula.

Constructing the Triangle with Angle 2x

By constructing the triangle with angle 2x, we need to find the length of the side opposite 2x. This can be derived from the properties of the right triangle formed by x.

The height of the triangle corresponding to angle 2x can be derived by considering the expansion of the original triangle. The height can be expressed as:

(2a cos x)

This is derived from the fact that doubling the angle effectively creates a larger triangle with the same height but a wider base.

Final Expression

Therefore, the expression for sin2x can be written as:

sin2x (opposite side) / (hypotenuse) (2a * b / c) / c

Substituting a c sin x and b c cos x into the expression:

sin2x 2 sin x cos x

Conclusion

We have shown that the identity:

sin2x 2 sin x cos x

Is true using the geometric properties of right triangles and the definitions of sine and cosine. This proof leverages the relationships between angles and the lengths of the sides opposite those angles without resorting to complex formulas or coordinate systems.

This method provides a clear and intuitive understanding of the trigonometric identity, making it accessible through basic geometric principles.