Trigonometric Functions: Angles and Degree Measure

Trigonometry is a branch of mathematics dealing with the relationships between the sides and angles of triangles.
This lecture begins with foundational concepts, specifically angles and degree measure, before delving into trigonometric functions and their applications.

Trigonometry has a wide array of practical applications across various fields:
- GPS and Navigation: Essential for positioning and mapping.
- Biorhythms: Used in the study and modeling of biological cycles.
- Propagation of Light and Sound: Models how waves travel.
- Engineering: Calculates forces on structures, such as bridges.
Ferris Wheel Example:
- Illustrates a direct application of trigonometric concepts to model real-world phenomena.
- A rider starts at the bottom of the Ferris wheel.
- As the wheel rotates, the rider's height from the ground changes.
- Plotting height against time on a Cartesian plane (horizontal axis = time, vertical axis = height) produces a wave-like graph.
- This waveform demonstrates how height oscillates up and down and repeats as the rotation continues.
- These wave models are also seen in light and sound propagation.

An angle is formed by connecting two rays (or half-lines) at a common point.
The shared point where the rays meet is called the vertex.
One ray is designated as the initial side, and the other as the terminal side.
Angles are conceptualized by starting the terminal side on top of the initial side and then rotating the terminal side. The amount of rotation creates the angle.
Angles are typically denoted using lowercase Greek letters, such as $\alpha$ , $\beta$ , $\gamma$ , and $\theta$ .

An angle is in standard position if:
- Its vertex is at the origin $(0,0)$ of a rectangular coordinate system.
- Its initial side lies along the positive x-axis.

The direction of rotation determines the sign of the angle measure:
- Counterclockwise Rotation: Results in a positive angle measure.
  - Example: An angle $\theta$ in standard position rotates counterclockwise, placing its terminal side in Quadrant II, indicating a positive measure (Figure A).
- Clockwise Rotation: Results in a negative angle measure.
  - Example: An angle $\theta$ in standard position rotates clockwise, placing its terminal side in Quadrant III, indicating a negative measure (Figure B).
The amount of rotation is crucial for determining the angle's magnitude:
- Figure A shows angle $\alpha$ created by a counterclockwise rotation (positive measure).
- Figure B shows angle $\beta$ created by a clockwise rotation (negative measure), even if the initial and terminal sides appear in the same 'position' as alpha. This highlights that rotation direction matters.
- Figure C shows angle $\gamma$ created by a counterclockwise rotation that includes a full revolution plus additional rotation, also having a positive measure.

Degrees are one of the two commonly used units for measuring angles (the other being radians, which will be discussed later).
Definition of One Degree: The angle formed by rotating an initial side exactly once in the counterclockwise direction until it coincides with itself measures $360^\circ$ (or three hundred and sixty degrees).
Key Angle Measures:
- One Full Revolution: A counterclockwise rotation where the terminal side returns to the initial side (positive x-axis) measures $360^\circ$ (Figure A). This is a ' $360$ ' in popular culture, like in skateboarding or gymnastics.
- One-Quarter Revolution: A counterclockwise rotation where the terminal side lands on the positive y-axis measures $90^\circ$ (Figure B). This is commonly known as a right angle and is denoted by a small square symbol at the vertex.
- One-Half Revolution: A counterclockwise rotation where the terminal side lands on the negative x-axis measures $180^\circ$ (Figure C). This is also referred to as a straight angle.

To sketch an angle in standard position:
1. Draw an x-y coordinate plane.
2. Place the vertex at the origin $(0,0)$ .
3. Draw the initial side along the positive x-axis.
4. Rotate the terminal side the specified amount and direction.
Example A: Draw an angle of $315^\circ$
- Rotate counterclockwise: $3 \times 90^\circ = 270^\circ$ first, then an additional $45^\circ$ .
- Total rotation: $270^\circ + 45^\circ = 315^\circ$ .
- The terminal side lies in Quadrant IV.
Example B: Draw an angle of $210^\circ$
- Rotate counterclockwise: $180^\circ$ (half revolution) first, then an additional $30^\circ$ .
- Total rotation: $180^\circ + 30^\circ = 210^\circ$ .
- The terminal side lies in Quadrant III.
Example C: Draw an angle of $-450^\circ$
- Rotate clockwise: One full clockwise revolution is $-360^\circ$ .
- Then, an additional clockwise rotation of $-90^\circ$ .
- Total rotation: $-360^\circ + (-90^\circ) = -450^\circ$ .
- The terminal side lies on the negative y-axis.
Example D: Draw an angle of $-45^\circ$
- Rotate clockwise: One-half of a right angle ( $90^\circ / 2 = 45^\circ$ ) in the clockwise direction.
- The terminal side lies in Quadrant IV.
Coterminal Angles Observation: The angles $315^\circ$ and $-45^\circ$ both have terminal sides that lie in the same position (in Quadrant IV), but they are described using different measures due to the direction and amount of rotation. This concept is fundamental to understanding coterminal angles.