The aim of this question is to find **sin t, cos t**, and **tan t** for a given point **P=(x,y)** on the unit circle which is determined by **t**. For this, we will be utilizing the **Cartesian Coordinate system** and **Equation of Circle**.

The basic concept behind this question is the knowledge of **the circle** and its **Coordinates in the Cartesian Coordinate System.** First, we will explain the concept of **Circle**, its **Equation**, and its **Coordinates in the Cartesian Coordinate System**.

A** Circle** is defined as a $2D$ geometrical structure have a constant radius $r$ across all two dimensions and its center point is fixed. Therefore, the **equation of a circle** is being derived by considering the position coordinates of circle centers with their constant radius $r$

\[{(x-a)}^2+{(y-b)}^2= r^2\]

This is the **Equation of the circle** where

$Center = A(a, b)$

$Radius = r$

For a **Standard Circle** in standard form, we know that the center has coordinates as $O(0,0)$ with $P(x,y)$ being any point on the sphere.

\[A(a, b) = O(0, 0)\]

By substituting the coordinates of the center in the above equation, we get:

\[{(x-0)}^2+{(y-0)}^2= r^2\]

\[x^2+y^2= r^2\]

Where:

\[x=r\ \cos \theta\]

\[y=r\ \sin \theta\]

## Expert Answer

Given in the question statement, we have:

Point $P(x, y)$ on the circle

Unit circle determined by $t$

We know that in the circle **x-coordinate** on the unit circle is cos $x= cos\ \theta$

So based on what is given here, it will be:

\[x=\cos t \]

We also know that in the circle **y-coordinate** on the unit circle is sin $y= \sin \theta$

So based on what is given here, it will be:

\[ y=\sin t\]

Thus we can say that:

\[ \tan \theta = \dfrac{\sin \theta}{\cos \theta}\]

Here it will be:

\[ \tan t = \dfrac{\sin t}{\cos t}\]

Putting values of $sin\ t = y$ and $cos\ t = x$ in the above equation, we get:

\[ \tan t = \dfrac{y}{x}\]

So the value of $tan\ t$ will be:

\[\tan t = \frac{y}{x}\]

## Numerical Results

The values of **$sin\ t$, $cos\ t$** and **$tan\ t$** for given point **$P=(x, y)$** on the unit circle which is determined by $t$ are as follows:

\[ \cos t = x \]

\[ \sin t = y\]

\[\tan t = \frac{y}{x}\]

## Example

If the terminal point determined by $t$ is $\dfrac{3}{5} , \dfrac{-4}{5}$ then calculate the values of **$sin\ t$, $cos\ t$** and **$tan\ t$** on the unit circle which is determined by $t$.

Solution:

We know that in the circle x-coordinate on the unit circle is cos $x= \cos\ \theta$

So based on what is given here, it will be:

\[x= \cos t \]

\[\cos t =\dfrac{3}{5}\]

We also know that in the circle y-coordinate on the unit circle is sin $y= \sin\ \theta$

So based on what is given here, it will be:

\[y= \sin t\]

\[\sin t=\dfrac{-4}{5}\]

Thus we can say that:

\[\tan t =\dfrac{\sin t}{\cos t}\]

\[\tan t =\dfrac{\dfrac{-4}{5}}{\dfrac{3}{5}}\]

So the value of $tan\ t$

\[\tan t = \dfrac{-4}{3}\]