Example: Show that the point \( P\left(\frac{3}{5},-\frac{4}{5}\right) \) is on the unit circle and state in which Quadrant it is in.

To show that the point \( P\left(\frac{3}{5},-\frac{4}{5}\right) \) is on the unit circle, we need to verify if it satisfies the equation of the unit circle, which is \( x^2 + y^2 = 1 \). Plugging in the coordinates, we have: \[ \left(\frac{3}{5}\right)^2 + \left(-\frac{4}{5}\right)^2 = \frac{9}{25} + \frac{16}{25} = \frac{25}{25} = 1. \] Since this holds true, point \( P \) lies on the unit circle. As for the coordinates, \( \frac{3}{5} \) is positive and \( -\frac{4}{5} \) is negative, which places point \( P \) in the fourth quadrant. In the fourth quadrant, x-values are positive while y-values are negative, and it’s where you find both the cosine values being positive and sine values being negative, reflecting rightward and downward directions respectively. So, not only is \( P \) on the unit circle, but it also embodies the distinctive character of the fourth quadrant. Additionally, the unit circle is quite pivotal in both trigonometry and the broader context of periodic functions, where points corresponding to angles translate into cosine and sine functions, showcasing their significance in wave patterns, oscillations, and even synchronization in physics. Understanding these positions can help in predicting motion and behaviors in various phenomena, like sound waves or electrical signals.