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(755) Exact Trig Ratios for Angles Greater than 90

Overview of Trigonometry with Larger Angles

  • Introduction to using special triangles and reference angles to find trigonometric ratios for angles greater than 90°.

  • Expansion of using sine, cosine, and tangent beyond the first quadrant.

Understanding Angles

  • Cartesian Grid: Visual reference for angles where:

    • Initial Arm: Positive x-axis.

    • Terminal Arm: Rotates counterclockwise (positive angle) or clockwise (negative angle) from the initial arm.

  • Angle (Theta): The angle between the terminal arm and the initial arm.

Principal Angles vs. Reference Angles

  • Principal Angle: The entire angle θ measured from the initial arm (counterclockwise).

  • Reference Angle (β): The acute angle between the terminal arm and the closest x-axis. Important for calculating trig ratios for obtuse angles.

Finding Reference Angles

  • Examples:

    • 250°: Reference angle = 250° - 180° = 70°.

    • 120°: Reference angle = 180° - 120° = 60°.

    • 300°: Reference angle = 360° - 300° = 60°.

    • 225°: Reference angle = 225° - 180° = 45°.

Trigonometric Ratios Using Reference Angles

  • When the terminal arm intersects a circle at point (x, y), the radius (r) acts as the hypotenuse of the right triangle formed.

  • Trigonometric Ratios:

    • Sine (sin θ) = Opposite / Hypotenuse = y / r

    • Cosine (cos θ) = Adjacent / Hypotenuse = x / r

    • Tangent (tan θ) = Opposite / Adjacent = y / x

CAST Rule for Quadrant Ratios

  • Quadrants Description:

    • 1st Quadrant: All ratios are positive.

    • 2nd Quadrant: Only sine is positive.

    • 3rd Quadrant: Only tangent is positive.

    • 4th Quadrant: Only cosine is positive.

  • Reasoning: Based on the signs of x and y coordinates in each quadrant, the CAST rule helps determine which trig functions are positive.

Special Triangles and Exact Values

  • Importance of special triangles for finding exact trig values.

  • Example: Sine 45° = 1/√2, which can be rationalized to √2/2.

Finding Exact Values for Larger Angles

  1. Sine of 210°:

    • Reference Angle: 30°

    • Using CAST, since we are in quadrant 3 (only tangent positive), sin(210°) = -sin(30°) = -1/2.

  2. Cosine of 240°:

    • Reference Angle: 60°

    • In quadrant 3, cos(240°) = -cos(60°) = -1/2.

  3. Tangent of 315°:

    • Reference Angle: 45°

    • In quadrant 4, tan(315°) = -tan(45°) = -1.

Trig Ratios from Points on Terminal Arm

  • Example: (5, -12) for angle Theta:

    • Triangle sides: 5 (adjacent), 12 (opposite), and hypotenuse from Pythagorean theorem = 13.

    • Ratios:

      • Sine(Theta) = -12/13

      • Cosine(Theta) = 5/13

      • Tangent(Theta) = -12/5.

  • Example: (8, 3) for angle Theta:

    • Triangle sides: 8 (adjacent), 3 (opposite), hypotenuse = √73.

    • Ratios:

      • Sine(Theta) = 3/√73

      • Cosine(Theta) = -8/√73

      • Tangent(Theta) = -3/8.

Unit Circle Overview

  • Unit Circle: A circle with a radius of 1.

    • Simplifies finding sine and cosine since coordinates directly give ratios.

    • Example: sine(270°) = -1; cosine(360°) = 1.

Co-terminal Angles

  • Co-terminal angles share the same terminal arm.

  • Example: 60°, 420°, 780° are co-terminal.

  • Negative angles example: -60° co-terminal with 300°.

Evaluating Trig Ratios with Negative Angles

  • Example: sine(-45°):

    • Reference angle = 45°. sine(-45°) is negative in quadrant 4, so answer = -1/√2, rationalized to √2/2.

  • Example: cosine(-60°) is co-terminal with 300°, giving positive ratio cosine(300°) = 1/2.