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.
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 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.
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°.
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
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.
Importance of special triangles for finding exact trig values.
Example: Sine 45° = 1/√2, which can be rationalized to √2/2.
Sine of 210°:
Reference Angle: 30°
Using CAST, since we are in quadrant 3 (only tangent positive), sin(210°) = -sin(30°) = -1/2.
Cosine of 240°:
Reference Angle: 60°
In quadrant 3, cos(240°) = -cos(60°) = -1/2.
Tangent of 315°:
Reference Angle: 45°
In quadrant 4, tan(315°) = -tan(45°) = -1.
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: 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 share the same terminal arm.
Example: 60°, 420°, 780° are co-terminal.
Negative angles example: -60° co-terminal with 300°.
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.