Exercise 2 Page 477

We are asked to use the unit circle to define the trigonometric functions of an angle. We will assume that the angle $\theta$ is in standard position. Let's begin by drawing the unit circle!

Next, we will draw an angle in standard position.

Let $x$ and $y$ respectively be the $x\text{-}$coordinate and $y\text{-}$coordinate of the intersection of the terminal side and the unit circle.

We can draw a right triangle using the same angle and the intersection. Since the circle is a unit circle, the hypotenuse of the right triangle is $1.$

We will now review the trigonometric functions.

Trigonometric Functions

$$\begin{array}{rlr}& \text{Sine}& \sin \theta =\frac{\text{Opposite side}}{\text{Hypotenuse}}\\ & \text{Cosine}& \cos \theta =\frac{\text{Adjacent side}}{\text{Hypotenuse}}\\ & \text{Tangent}& \tan \theta =\frac{\text{Opposite side}}{\text{Adjacent side}}\\ & \text{Cotangent}& \cot \theta =\frac{\text{Adjacent side}}{\text{Opposite side}}\\ & \text{Secant}& \sec \theta =\frac{\text{Hypotenuse}}{\text{Adjacent side}}\\ & \text{Cosecant}& \csc \theta =\frac{\text{Hypotenuse}}{\text{Opposite side}}\end{array}$$

Looking back at our diagram, we find the length of each side. Finally, we will write the trigonometric functions in terms of $x$ and $y.$ Let's do it!