3.3 Trigonometric Functions of Any Angle

Trigonometric functions extend beyond the basic right triangle, applying to all angles in the unit circle. This broader view allows us to work with angles of any size, even those greater than 360° or negative angles.

Understanding how trig functions behave in different quadrants is crucial. The ASTC rule helps remember which functions are positive where, while reference angles simplify calculations for non-standard angles. These concepts are key to mastering trigonometry.

Understanding Trigonometric Functions in All Quadrants

Trigonometric functions in all quadrants

• Unit circle defines trig functions for all real numbers allows angles > 360° or < 0° (full rotations)
• Coterminal angles share terminal side differ by multiples of 360° ($720°$ and $0°$)
• Periodic nature repeats every 360° (2π radians) ($\sin \theta = \sin (\theta + 360°)$)
• Quadrant-specific characteristics unique properties for trig functions (Q1: all positive, Q2: only sine positive)

Signs of trigonometric functions

• ASTC rule All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4
• Reciprocal functions follow primary functions (cosecant follows sine)
• x and y coordinates sine relates to y-coordinate cosine to x-coordinate
• Tangent positive when sine and cosine have same sign negative when opposite ($\tan 45° > 0$, $\tan 225° < 0$)

Reference angles for evaluation

• Reference angle acute angle with x-axis always positive ≤ 90°
• Calculating reference angles:
  1. Q1: θ
  2. Q2: 180° - θ
  3. Q3: θ - 180°
  4. Q4: 360° - θ
• Using reference angles absolute value equals reference angle apply quadrant sign ($\sin 150° = \sin 30°$)

Unit circle for problem-solving

• Unit circle radius 1 unit center at origin (0, 0)
• Coordinates x = cosθ, y = sinθ
• Special angles 30°, 45°, 60° and multiples memorize for efficiency ($\cos 30° = \frac{\sqrt{3}}{2}$)
• Angle measure conversions $θ_{rad} = θ_{deg} × \frac{π}{180°}$, $θ_{deg} = θ_{rad} × \frac{180°}{π}$
• Solving triangles find missing sides or angles using unit circle values
• Graphing trig functions plot key points using unit circle ($\sin 90° = 1$, $\cos 180° = -1$)