Step-by-Step Solution

Function Plots

Trigonometric Functions

The six basic trigonometric functions are:

Primary Functions

• \[ \sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} \]
• \[ \cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
• \[ \tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(x)}{\cos(x)} \]

Reciprocal Functions

• \[ \csc(x) = \frac{1}{\sin(x)} = \frac{\text{hypotenuse}}{\text{opposite}} \]
• \[ \sec(x) = \frac{1}{\cos(x)} = \frac{\text{hypotenuse}}{\text{adjacent}} \]
• \[ \cot(x) = \frac{1}{\tan(x)} = \frac{\text{adjacent}}{\text{opposite}} = \frac{\cos(x)}{\sin(x)} \]

Properties

• Domain and Range:
  • sin(x), cos(x): Domain = All real numbers, Range = [-1, 1]
  • tan(x), cot(x): Domain = All real numbers except undefined points, Range = All real numbers
  • sec(x), csc(x): Domain = All real numbers except undefined points, Range = (-∞, -1] ∪ [1, ∞)
• Periodicity:
  • sin(x), cos(x), sec(x), csc(x): Period = 2π (360°)
  • tan(x), cot(x): Period = π (180°)
• Symmetry:
  • Odd functions: sin(x), tan(x), cot(x), csc(x)
  • Even functions: cos(x), sec(x)

Common Values

• \[ \sin(0°) = 0, \cos(0°) = 1, \tan(0°) = 0 \]
• \[ \sin(30°) = \frac{1}{2}, \cos(30°) = \frac{\sqrt{3}}{2}, \tan(30°) = \frac{1}{\sqrt{3}} \]
• \[ \sin(45°) = \frac{1}{\sqrt{2}}, \cos(45°) = \frac{1}{\sqrt{2}}, \tan(45°) = 1 \]
• \[ \sin(60°) = \frac{\sqrt{3}}{2}, \cos(60°) = \frac{1}{2}, \tan(60°) = \sqrt{3} \]
• \[ \sin(90°) = 1, \cos(90°) = 0, \tan(90°) = \text{undefined} \]

Applications

• Trigonometry
• Calculus
• Physics (wave motion)
• Engineering (signal processing)
• Navigation and surveying
• Music and sound analysis