91Ó°ÊÓ

The tangent function, often written as \( \tan \theta \), is one of the primary trigonometric functions. It relates an angle in a right triangle to the ratio of the opposite side's length to that of the adjacent side. In the context of the unit circle, which is a circle with a radius of one, the tangent of an angle \( \theta \) is the length of the line segment from the origin to the point where the line containing the terminal side of \( \theta \) intersects the tangent line to the circle at \( (1,0) \).

It is worth mentioning that the tangent function is periodic, which means it repeats its values in regular intervals. Specifically, the period of the tangent function is \( \pi \) radians, or \( 180^{\circ} \). This means the function will have the same value every \( \pi \) radians.

Notably, the tangent function has vertical asymptotes, which occur at odd multiples of \( \frac{\pi}{2} \) radians. This is because tangent approaches infinity as it nears these points. Consequently, \( \tan \theta \) is undefined at these angles.

The range of the tangent function is all real numbers. Unlike the sine and cosine functions, which are limited to functioning within the interval [\(-1,1\)], the tangent function can take on any value from \(-\infty\) to \(+\infty\).

Unlike sine and cosine, which are continuous throughout their range of \([0, 2\pi]\), the tangent function has breaks (asymptotes) at every odd multiple of \( \frac{\pi}{2} \). These breaks are where the function switches from approaching positive infinity to negative infinity.

This characteristic allows the tangent function to have solutions for equations where the tangent value is any real number. Therefore, it is entirely possible for the equation \( \tan \theta = 4\sqrt{5} \) to have a solution, since \( 4\sqrt{5} \) is a real number within its range.

Real numbers are the union of both rational numbers, such as fractions and integers, and irrational numbers, like \( \sqrt{2} \) or \( \pi \). They can be positive, negative, or zero and are represented on a continuous number line without any gaps between them.

A valuable property of real numbers is that they are dense, meaning between any two real numbers, there lies an infinite set of other real numbers. This makes them incredibly comprehensive.

When considering the tangent function, the fact that its range is all real numbers helps when solving equations like \( \tan \theta = x \). Provided that \( x \) is a real number, there will be some angle \( \theta \) that satisfies such an equation. Therefore, when we assess whether "\( \tan \theta = 4\sqrt{5} \)" is possible, it indeed is because \( 4\sqrt{5} \) remains within the real number set.