Trigonometric functions depend solely on angles and do not rely on the radius of the defining circle. The independence from the radius means that one can easily calculate trigonometric function values using any convenient radius. For instance, for an angle of \frac{\pi}{3}, one could use a radius of 2, and for an angle of \frac{\pi}{4}, a radius of \sqrt{2} could be employed.
The fact that trigonometric functions do not depend on the radius provides the basis for considering any radius when calculating their values. This article will elucidate this point.
1 Trigonometric Function Values Are Independent of Radius
As illustrated in Figure 1, consider two circles of different sizes. The radius of the smaller circle is 1. A point A(x, y) corresponds to the angle \theta on the circumference of this circle. Next, we multiply the radius of this circle by r (r > 0), where r is any positive real number. We shall create a circle with this radius r and consider the point A' as depicted in Figure 1.
The two orange triangles within Figure 1 are similar, as they scale the hypotenuse by a factor of r while keeping \theta fixed. Thus, the coordinates of point A' become (rx, ry).
Now, let us calculate the trigonometric function values for points A(x, y) and A'(rx, ry) using the definition of trigonometric functions.
Calculating Trigonometric Functions Using A(x, y):
\begin{aligned} \sin \theta &= \frac{y}{1} = y,\\ \cos \theta &= \frac{x}{1} = x,\\ \tan \theta &= \frac{y}{x}. \end{aligned} \tag{1}
Calculating Trigonometric Functions Using A'(rx, ry):
\begin{aligned} \sin \theta &= \frac{ry}{r} = y,\\ \cos \theta &= \frac{rx}{r} = x,\\ \tan \theta &= \frac{ry}{rx} = \frac{y}{x}. \end{aligned} \tag{2}
By comparing Equation 1 and Equation 2, we can indeed verify that the values of the trigonometric functions coincide. This demonstrates that the values of trigonometric functions are independent of the radius, meaning that if \theta is the same, the resulting values will be identical.
2 Conclusion
This article has shown that trigonometric functions do not depend on the radius of the defining circle. This fact supports the notion that when calculating trigonometric functions for representative values such as \frac{\pi}{3}, any radius can be utilised.
