Trigonometry

1 Angles

  • An angle is formed by rotating a ray around a fixed point (vertex).

    • Initial side: starting position of the ray.

    • Terminal side: position after rotation.

  • 1.1 Standard Position:

    • Vertex at origin, initial side along positive x-axis.

  • 1.2 Positive and Negative Angles:

    • Positive angle: measured counterclockwise.

    • Negative angle: measured clockwise.

1.3 Co-terminal Angles

  • Angles that share the same initial and terminal sides but differ by full rotations (360° or 2π radians).

    • Positive Co-terminal Angles: Angle + 360° (e.g., for 45°, 405°).

    • Negative Co-terminal Angles: Angle - 360° (e.g., for 45°, -315°).

  • Co-terminal angles represented as θ + 360°n or θ - 360°n, where n ∈ Z.

1.4 Quadrants

  • Cartesian plane split into four quadrants:

    • Quadrant 1: 0° to 90°

    • Quadrant 2: 90° to 180°

    • Quadrant 3: 180° to 270°

    • Quadrant 4: 270° to 360°

1.5 Introduction to Angle Types

  • Types of Angles:

    • Acute Angle: 0° < θ < 90°.

    • Right Angle: θ = 90°.

    • Obtuse Angle: 90° < θ < 180°.

    • Straight Angle: θ = 180°.

    • Reflex Angle: 180° < θ < 360°.

    • Full Angle: θ = 360°.

    • Quadrantal Angle: Held on axes (0°, 90°, 180°, 270°).

1.6 Complementary and Supplementary Angles

  • Complementary Angles: Sum equals 90° (e.g., 30° and 60°).

  • Supplementary Angles: Sum equals 180° (e.g., 110° and 70°).

1.7 Angle Properties Related to Parallel Lines

  • Corresponding Angles: Same relative position at each intersection.

  • Alternate Angles: Lie opposite sides of a transversal and are equal.

  • Interior Angles: Supplementary (sum equals 180°).

  • Vertically Opposite Angles: Equal in measure when lines intersect.

1.8 Units for Angle Measurements

  • Degree: Commonly used unit; 1° = 1/360 of full circle.

  • Radian: Standard unit in math and science; 1 radian= angle subtended when arc length = radius.

  • Relationship between Degrees and Radians:

    • 360° = 2π rad, or 180° = π rad.

  • Conversion:

    • From degrees to radians: Multiply degrees by π/180.

    • From radians to degrees: Multiply radians by 180/π.

2 Arc Length and Sector Area of a Circle

  • Arc: Part of the circumference; minor and major arcs.

  • Sector: Area enclosed between two radii.

  • Formulas:

    • Arc Length L = rθ (θ in radians).

    • Area of Sector A = (1/2)r²θ.

Example Problems

  • Various examples illustrating concepts including co-terminal angles, angle properties, and sectors

  • Problems include finding arc lengths, sector areas, and understanding angle types in different scenarios.