**What we learn in this section:**

### Radian Measure

**radian**measure. The use of this unit simplifies many formulas in trigonometry and calculus.

*unit*circle (i.e. circle of radius 1). This angle cuts out an arc on the circumference of this circle. The length of this arc is called the size of the angle in radians (Figure 1). Because the total circumference of the unit circle (note that the radius is 1) is 2 π and one complete rotation is 360○ , we have

\[1^{\circ}=\frac{\pi}{180}\text{ radians}\approx0.017\text{ radians.}\]

In mathematics, whenever angles occur, the default is that the angles are measured in radians (instead of degrees) unless you see the degree symbol (^{○}). Therefore if you want to calculate sin 1, you need to set your calculator in radian mode and if you want to find sin 1^{○}, you need to set your calculator in degree mode. Therefore

and when we talk about the angle $\pi/6$, we mean $\pi/6$ radians (which is 30○ ) not $\pi/6$ ( $\approx$ 0.524 ) degrees.

Figure 1 |

### Arc Length

An arc on a circle can be measured by the angle it subtends in radians. What is the length *s* of this arc, as in Figure 2? The answer is as follows.

**Theorem 1 (Arc Length):**If $s$ is the length of an arc of $\theta$ radians on a circle of

radius $r$, then

\[

s=r\theta\qquad(\theta\text{ in radians)}.

\]

Figure 2: An arc intercepted by the central angle $\theta$. |

**sector**(or simply a sector) is a portion of a disk enclosed by two radii and an arc (the shaded part of Figure 3):

Figure 3: A circular sector is shaded in purple. |

**Theorem 2 (Area of a Sector):**The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ is given by:\[A=\frac{1}{2}r^2\theta\]where $\theta$ is measured in radians.

- The unit of $A$ is the square of the length unit used to measure $r$. For example, if $r$ is measured in meters, then $A$ will be measured in the square meters.

### Directed Angles

Any angle has two sides. To measure rotations, we designate one side as the **initial side** and the other as the **terminal side**. Then we think of angle as being generated by the rotation of the initial side to the final side. If the rotation is **counterclockwise**, the angle is regarded **positive** and if the rotation is **clockwise**, the angle is **negative** (Figure 4).

- We allow angles of more than one complete rotation. That is, we also allow $\theta>2\pi$ or $\theta<-2\pi$.

### Angles in standard position

We say the angle is in its **standard position** if its vertex is at the origin of the coordinate system and its initial side is along the positive $x$-axis.

(a) A positive angle in standard position. | (b) A negative angle in standard position |

**Figure 4**

We say an angle is a **first quadrant angle** or is **in the first quadrant** if in standard position, its terminal end lies in the first quadrant. Similar definitions hold for second, third or fourth quadrant angles (Figure 5(a)). For example, $\pi/4, -315^\circ$and $405^{\circ}$ are first quadrant angles (Figure 5(b)), $5\pi/6$ is a second quadrant angle (Figure 5(c)), and $-60^{\circ}$ and $300^{\circ}$ are fourth quadrant angles.

(a) The angle shown here is a second quadrant angle | (b) The angles shown here are first quadrant angles |

(c) The angle shown here is a second quadrant angle | (d) The angles shown here are fourth quadrant angles |

**Figure 5**