Precalculus 4.1

Introduction to Angles and Trigonometry

  • Today's objective: Introduction to trigonometry, angular measurement, degrees and radians, circular arc length.

  • Angles are the domain elements of trigonometric functions.

Angular Measurement

  • Initial Side: The starting position of an angle, akin to facing east on a compass.

  • Terminal Side: The final position of the angle after rotation.

  • Positive Angles: Rotated counterclockwise (e.g., 45°).

  • Negative Angles: Rotated clockwise (e.g., -45°).

Standard Position of Angles

  • An angle in standard position has its initial side along the positive x-axis and its terminal side can open either counterclockwise or clockwise.

  • Example Angles:

    • Positive Angle: 60° (counterclockwise)

    • Negative Angle: -225° (clockwise)

    • Coterminal Angles: Angles that share the same terminal side (e.g., -225° and 135°).

    • Calculation of Coterminal Angles: Infinite coterminal angles can be calculated by adding or subtracting multiples of 360°.

Angular Measurement in Degrees

  • Why 360 Degrees?:

    • Base 10 is common in everyday measurements; however, 360 was chosen because it has more integer factors than 100, allowing easier divisions for practical applications like time (60 minutes in an hour).

Introduction to Radians

  • Radian Measurement: A new angle measurement based on properties of a circle.

  • Definition of Radian: The angle subtended by an arc length equal to the radius of a circle.

  • Circle Properties: The full circumference of a circle is equal to 2π radians.

Conversion Between Degrees and Radians

  • Degrees to Radians: Multiply by π/180.

  • Radians to Degrees: Multiply by 180/π.

  • Equivalents: Common radian degrees to remember:

    • 360° = 2π

    • 180° = π

    • 90° = π/2

    • 60° = π/3

    • 45° = π/4

    • 30° = π/6

Arc Length Calculation

  • Arc Length Formula:

    • For a circle with radius r and angle θ in radians, arc length s = r * θ.

  • Example Problem:

    • Length of an arc for 1/4 radian in a circle of radius 3 inches is calculated by:

      • s = 3 * (1/4) = 0.75 inches.

Problem-Solving Example: Pizza Safety Slice

  • To find the perimeter of a 30° slice of an 8-inch pizza:

    • Arc length calculation gives 4/3 inches.

    • Total perimeter = 8 + 8 + (4/3) = 16 + 4/3.

Navigation and Angle Measurements

  • Navigation Adjustments: In navigation, the initial side of angles is considered to face north, with positive angles moving clockwise instead of counterclockwise.

  • Bearings as measurements: Refers to angles measured clockwise from due north, often applied in navigation scenarios (e.g., travel directions).

Conclusion

  • Understanding angles in both standard position and navigational context is vital for a solid foundation in trigonometry.