# MTH 111–112 Supplement

## Section2.2Generalized Definitions of Trigonometric Functions

We can generalize the definitions of the trigonometric functions so that they are applicable to circles of any size.

### Definition2.2.1.

If the point $$P = \left(x,y \right)$$ is specified by the angle $$\theta$$ on the circumeference of a circle of radius $$r\text{,}$$ then $$\cos (\theta ) = \frac{x}{r}$$ and $$\sin (\theta ) = \frac{y}{r} \text{.}$$
Notice that if we are on a unit circle, where $$r=1\text{,}$$ then these definitions for $$\cos(\theta)$$ and $$\sin(\theta)$$ simplifly accordingly:
\begin{equation*} \cos(\theta)=\frac{x}{r}=\frac{x}{1}=x \end{equation*}
\begin{equation*} \sin(\theta)=\frac{y}{r}=\frac{y}{1}=y \end{equation*}
We can use Definition 2.2.1 to express the other four trigonometric functions in terms of $$x\text{,}$$ $$y\text{,}$$ and $$r\text{.}$$
\begin{align*} \tan(\theta) \amp= \frac{\sin(\theta)}{\cos(\theta)} \\ \amp= \frac{y/r}{x/r} \\ \amp= \frac{y}{x} \end{align*}
\begin{align*} \csc(\theta) \amp= \frac{1}{\sin(\theta)} \\ \amp= \frac{1}{y/r} \\ \amp= \frac{r}{y} \end{align*}
\begin{align*} \sec(\theta) \amp= \frac{1}{\cos(\theta)} \\ \amp= \frac{1}{x/r} \\ \amp= \frac{r}{x} \end{align*}
\begin{align*} \cot(\theta) \amp= \frac{1}{\tan(\theta)} \\ \amp= \frac{1}{y/x} \\ \amp= \frac{x}{y} \end{align*}
We summarize this in the following definition.

### Definition2.2.3.

If the point $$P = \left(x,y \right)$$ is specified by the angle $$\theta$$ on the circumeference of a circle of radius $$r\text{,}$$ then we can define the six trigonemtric functions as follows.
\begin{equation*} \cos (\theta ) = \frac{x}{r} \end{equation*}
\begin{equation*} \sin (\theta ) = \frac{y}{r} \end{equation*}
\begin{equation*} \tan (\theta ) = \frac{y}{x} \end{equation*}
\begin{equation*} \sec (\theta ) = \frac{r}{x} \end{equation*}
\begin{equation*} \csc (\theta ) = \frac{r}{y} \end{equation*}
\begin{equation*} \cot (\theta ) = \frac{x}{y} \end{equation*}

### Example2.2.5.

Find the exact value of each of the six trigonometric functions of an angle $$\theta$$ if $$(-1,2)$$ is a point on its terminal side.
Solution.
We need the values of $$x\text{,}$$ $$y\text{,}$$ and $$r$$ to determine the exact value of each of the trigonemtric function. We are given $$x$$ and $$y\text{,}$$ but will need to find the value of $$r\text{.}$$
We can think of $$r$$ as the hypotenuse of a right triangle whose horiztonal leg has a length of $$|-1|=1$$ unit and vertical leg has a length of $$2$$ units. We can use the Pythagorean Theorem to solve for $$r\text{:}$$
\begin{align*} r^2 \amp= 1^2+2^2 \\ r^2 \amp= 5 \\ r \amp= \sqrt{5} \end{align*}
Now that we have values for $$x\text{,}$$ $$y\text{,}$$ and $$r\text{,}$$ we can use Definition 2.2.3 to state the exact values of each trigonmetric function.
\begin{align*} \cos(\theta) \amp= \frac{x}{r} \\ \amp= \frac{-1}{\sqrt{5}} \\ \amp= \frac{-1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \\ \amp= -\frac{\sqrt{5}}{5} \end{align*}
\begin{align*} \sin(\theta) \amp= \frac{y}{r} \\ \amp= \frac{2}{\sqrt{5}} \\ \amp= \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \\ \amp= \frac{2\sqrt{5}}{5} \end{align*}
\begin{align*} \tan(\theta) \amp= \frac{y}{x} \\ \amp= \frac{2}{-1} \\ \amp= -2 \end{align*}
\begin{align*} \sec(\theta) \amp= \frac{r}{x} \\ \amp= \frac{\sqrt{5}}{-1} \\ \amp= -\sqrt{5} \end{align*}
\begin{align*} \csc(\theta) \amp= \frac{r}{y} \\ \amp= \frac{\sqrt{5}}{2} \end{align*}
\begin{align*} \cot(\theta) \amp= \frac{x}{y} \\ \amp= \frac{-1}{2} \\ \amp= -\frac{1}{2} \end{align*}
We can use Definition 2.2.1 to express the coordinates of point $$P$$ in Figure 2.2.2. We do this by solving the equations $$\cos (\theta ) = \frac{x}{r}$$ and $$\sin (\theta ) = \frac{y}{r}$$ for $$x$$ and $$y\text{,}$$ respectively:
\begin{equation*} \cos(\theta)=\frac{x}{r} \implies x=r\cos(\theta) \end{equation*}
\begin{equation*} \sin(\theta)=\frac{y}{r} \implies y=r\sin(\theta) \end{equation*}
We summarize this below.

### Definition2.2.7.

If the point $$P = \left(x,y \right)$$ is specified by the angle $$\theta$$ on the circumeference of a circle of radius $$r\text{,}$$ then $$x=r \cos (\theta )$$ and $$y=r \sin (\theta )\text{.}$$

### Example2.2.9.

Find the coordinates of the point on a circle with a radius of $$8$$ units corresponding to an angle of $$\frac{7\pi}{4}\text{.}$$
Solution.
We are given the values of $$r$$ and $$\theta\text{,}$$ so we can use Definition 2.2.7 to determine the coordinate values.
Find $$x\text{:}$$
\begin{align*} x \amp= r\cos\mathopen{}\left( \theta \right)\mathclose{} \\ \amp= 8\cos\mathopen{}\left( \frac{7\pi}{4}\right)\mathclose{} \\ \amp= 8 \cdot \frac{\sqrt{2}}{2} \\ \amp= 4\sqrt{2} \end{align*}
Find $$y\text{:}$$
\begin{align*} y \amp= r\sin\mathopen{}\left( \theta \right)\mathclose{} \\ \amp= 8\sin\mathopen{}\left( \frac{7\pi}{4}\right)\mathclose{} \\ \amp= 8 \mathopen{}\left( -\frac{\sqrt{2}}{2} \right)\mathclose{} \\ \amp= -4\sqrt{2} \end{align*}
So the coordinates of the point on a circle with a radius of $$8$$ units corresponding to an angle of $$\frac{7\pi}{4}$$ are $$\mathopen{}\left( 4\sqrt{2}, -4\sqrt{2} \right)\mathclose{}\text{.}$$

### ExercisesExercises

#### Six Trigonemtric Function Values.

In Exercises 1–2, find the exact value of each of the six trigonometric functions of an angle $$\theta$$ if the given point is on its terminal side.
##### 1.
$$(3,4)$$
##### 2.
$$(-2,-6)$$

#### Find the coordinates.

In Exercises 3–4, find the coordinates of the point on a circle with the given radius corresponding to the given angle.
##### 3.
$$r=3, \theta=\frac{7\pi}{6}$$
##### 4.
$$r=10, \theta=\frac{\pi}{3}$$