Question

From the information given, find the quadrant in which the terminal point is determined by $t$lies.

$\tan t>0\text{andsin}t<0$

Short answer

The quadrant in which the terminal point is determined by $t$lies isquadrant III.

Step-by-step solution

Step 1. State the definition of the terminal point on the unit circle.

Starting at the point $(1,0)$, if $t\ge 0$, then $t$is the distance along the unit circle in a clockwise direction and if $t<0$, then $\left|t\right|$ is the distance along the unit circle in a clockwise direction. Here, $t$ is a real number and this distance generates a point $P\left(x,y\right)$ on the unit circle. The point $P\left(x,y\right)$ obtained in this way is called the terminal point determined by the real number $t$.

Step 2. Find the sign of x and y coordinates.

Determine the signs of $x$and $y$coordinates by observing the signs of trigonometric functions.

Note that for the given point t on the unit circle $\cos t=x$, $\sin t=y$and $\tan t=\frac{y}{x}$where $\left(x,y\right)$is the terminal point to a real number $t$.

$$\begin{gathered} \sin t=y \\ <0 \end{gathered}$$

And,

$$\begin{gathered} \tan t=\frac{y}{x} \\ >0 \end{gathered}$$

But since $y<0$, therefore, to make $\tan t>0$, $x<0$.

Therefore, $x$-coordinate is negative and $y$-coordinate is also negative.

Step 3. Find the quadrant.

Since $x$-coordinate is negative and $y$-coordinate is also negative, it implies that the terminal point lies in quadrant III.