91Ó°ÊÓ

Trigonometric functions like sine, cosine, and tangent are periodic, which means their value repeats after a certain interval. For sine and cosine, this interval is \(2\pi\), whereas for the tangent, it is \(\pi\). The sine and cosine functions have a period of \(2\pi\) because they return to their initial values after rotating a full circle of \(360^\circ\) or \(2\pi\) radians. The tangent function has a period of \(\pi\) because it repeats its values after rotating half a circle or \(\pi\) radians.

When \(\theta\) equals an integer multiple of \(\frac{\pi}{2}\), sine and cosine functions have specific values. Specifically, the sine function takes on integer values of 0, 1, and -1, while the cosine function takes on integer values of 0 and 1. Considering sine and cosine are essential in the study of trigonometry, having \(\theta\) as an integer multiple of \(\frac{\pi}{2}\) would make the trigonometric functions less useful because they would not be continuous and would result in undefined values in some cases, such as the tangent function. For example, \(sin(\frac{\pi}{2}) = 1\), \(cos(\frac{\pi}{2}) = 0\), and \(tan(\frac{\pi}{2})\) is undefined.

The tangent function's domain excludes integer multiples of \(\frac{\pi}{2}\) because of the properties of the sine and cosine functions. The tangent function is defined as: \[tan(\theta) = \frac{sin(\theta)}{cos(\theta)}\] Since the cosine function is equal to 0 at every integer multiple of \(\frac{\pi}{2}\), the tangent function becomes undefined for those values of \(\theta\). Division by zero is not allowed in mathematics, and therefore, the values of \(\theta\) which make the denominator zero are excluded from its domain. In conclusion, integer multiples of \(\frac{\pi}{2}\) are excluded from the allowable values for \(\theta\) because they lead to undefined values for some trigonometric functions, such as the tangent function, and also make the properties of trigonometric functions less useful in the study of trigonometry.