3.0 Sign of trigonometric functions

Let us consider a plane which is divided into $4$ quadrants by $X$-axis and $Y$-axis such that value of $x$ varies as $$\begin{equation} \begin{aligned} I{\text{ Quadrant}} \equiv {\text{0}} < x < \frac{\pi }{2} \\ II{\text{ Quadrant}} \equiv \frac{\pi }{2} < x < \pi \\ III{\text{ Quadrant}} \equiv \pi < x < \frac{{3\pi }}{2} \\ IV{\text{ Quadrant}} \equiv \frac{{3\pi }}{2} < x < 2\pi \\\end{aligned} \end{equation} $$ Now, let us recall from previous topic, a circle of radius $1$ unit and centre at origin $O(0,0)$. Take four different points in each quadrant as shown in figure.

Since for every point on the circle of radius $1$ unit, $a$ and $b$ varies as $$ - 1 \leqslant a \leqslant 1,\; - 1 \leqslant b \leqslant 1$$From our previous discussion we assumed that $$\cos x = a,\;\sin x = b$$Therefore, we can say that $$\begin{equation} \begin{aligned} {\text{For }}I{\text{ Quadrant, both }}a{\text{ and }}b{\text{ are}} + ve \\ \Rightarrow \cos x = + ve,\;\sin x = + ve \\ {\text{For }}II{\text{ Quadrant, }}a{\text{ is}} - ve{\text{ and }}b{\text{ is }} + ve \\ \Rightarrow \cos x = - ve,\;\sin x = + ve \\ {\text{For }}III{\text{ Quadrant, }}a{\text{ is}} - ve{\text{ and }}b{\text{ is }} - ve \\ \Rightarrow \cos x = - ve,\;\sin x = - ve \\ {\text{For }}IV{\text{ Quadrant, }}a{\text{ is + }}ve{\text{ and }}b{\text{ is }} - ve \\ \Rightarrow \cos x = + ve,\;\sin x = - ve \\\end{aligned} \end{equation} $$ All other trigonometric functions can be written in terms of $sinx$ and $cosx$, therefore, their sign is also dependent on the sign of these two trigonometric functions as shown in table below.