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Let P(x,y) be the terminal point on the unit circle determined by t. Then find the value for sin(t), cos(t) and tan(t).

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 $tan\ t$

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

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