Welcome to Chapter 6, Lesson 1! This lesson marks the beginning of our exciting journey into Trigonometry. We are moving beyond basic geometry to understand how angles, circles, and coordinates interact on a deeper level. This week, we are laying the foundation with the Unit Circle and angle measurement—concepts that will serve as the building blocks for the rest of the course.

Key Topics Covered

Here is the roadmap for this lesson. We will be covering a wide range of fundamental skills:

  • Converting Degrees-Minutes-Seconds (DMS) to decimal degrees (and vice-versa)
  • Converting between Degree and Radian measures
  • Identifying Coterminal Angles
  • Calculating Arc Length and the Area of a Sector
  • Sketching angles in standard position
  • Navigating the Unit Circle: Finding coordinates and special angles
  • Evaluating trigonometric functions using reference angles

1. Angle Measurement and Conversions

In this lesson, we treat angles with more precision. You are used to decimal degrees (e.g., $45.5^{\circ}$), but we also use the DMS system where:

  • $1^{\circ} = 60'$ (60 minutes)
  • $1' = 60''$ (60 seconds)

We also introduce the Radian. It is crucial to remember the primary conversion factor:

$$180^{\circ} = \pi \text{ radians}$$

To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.

2. The Geometry of Circles

Once we understand radians, we can calculate properties of circles much easier. Remember, for the following formulas, $\theta$ must be in radians:

  • Arc Length ($s$): The distance along the curve of the circle. $$s = r\theta$$
  • Area of a Sector ($A$): The "slice of pizza" area. $$A = \frac{1}{2}r^2\theta$$

3. The Unit Circle and Special Angles

The Unit Circle is a circle with a radius of $r=1$ centered at the origin $(0,0)$. The equation for this circle is $x^2 + y^2 = 1$. This is the most powerful tool in trigonometry because it connects algebra to geometry.

Any point $(x, y)$ on the unit circle corresponds to an angle $\theta$ such that:

$$x = \cos(\theta) \quad \text{and} \quad y = \sin(\theta)$$

We will focus heavily on Special Angles (multiples of $30^{\circ}, 45^{\circ}, 60^{\circ}$ or $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$). Memorizing the first quadrant coordinates for these angles is essential!

4. Reference Angles

Finally, we solve problems using Reference Angles (${\theta}'$). The reference angle is always the positive, acute angle formed between the terminal side of your angle and the x-axis. This helps us find trigonometric values in all four quadrants by determining the sign ($+$ or $-$) based on the quadrant.

Professor Baker’s Tip: When sketching angles or finding coterminal angles, visualize the circle's rotation. Positive angles rotate counter-clockwise, and negative angles rotate clockwise. Keep practicing those conversions!