Question

The terminal point$P\left(x,y\right)$determined by a real number $t$ is given. Find$\sin t,\cos t$and$\tan t$.

$\left(\frac{40}{41},\frac{9}{41}\right)$

Short answer

$\sin t=\frac{9}{41},\cos t=\frac{40}{41}$, $\tan t=\frac{9}{40}$.

Step-by-step solution

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

Starting at the point $\left(1,0\right)$, if $t\ge 0$, then $t$ is the distance along the unit circle in clockwise direction and if $t<0$, then$\left|t\right|$ is the distance along the unit circle in 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 sint and cost.

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 real number $t$. Since $P\left(x,y\right)=\left(\frac{40}{41},\frac{9}{41}\right)$, therefore,$x=\frac{40}{41}$ and $y=\frac{9}{41}$.

So,

$$\begin{gathered} \sin t=y \\ =\frac{9}{41} \end{gathered}$$

And,

$$\begin{gathered} \cos t=x \\ =\frac{40}{41} \end{gathered}$$

Step 3. Find tant.

To find $\tan t$substitute $\frac{40}{41}$ for $x$ and $\frac{9}{41}$ for $y$ into the expression $\tan t=\frac{y}{x}$and simplify.

$$\begin{gathered} \tan t=\frac{y}{x} \\ =\frac{\left(\frac{9}{41}\right)}{\left(\frac{40}{41}\right)} \\ =\frac{9}{40} \end{gathered}$$

Hence, $\tan t=\frac{9}{40}$.