Trigonometry Flashcards

The Unit Circle

• Equation of the unit circle: x^2 + y^2 = 1
• Sine function: Input is \theta, output is the y-coordinate.
• Cosine function: Input is \theta, output is the x-coordinate.

Reference Angles

• Reference angle \theta': Acute angle formed by the terminal side of \theta and the horizontal axis.

The Six Trigonometric Functions

• Tangent function: Input is \theta, output is \frac{y}{x}.
• \tan \theta = \frac{\sin \theta}{\cos \theta}
• Reciprocal Identities:
  • Cosecant: Reciprocal of sine.
  • Secant: Reciprocal of cosine.
  • Cotangent: Reciprocal of tangent.

Right Triangle Trigonometry

• \sin \theta = \frac{y}{r}, \cos \theta = \frac{x}{r}, \tan \theta = \frac{y}{x} where r = \sqrt{x^2 + y^2}
• Using "opp," "adj," and "hyp" for a right triangle:
  • \sin \theta = \frac{opp}{hyp}, \csc \theta = \frac{hyp}{opp}
  • \cos \theta = \frac{adj}{hyp}, \sec \theta = \frac{hyp}{adj}
  • \tan \theta = \frac{opp}{adj}, \cot \theta = \frac{adj}{opp}

Trigonometric Identities

• Cofunctions of complementary angles are equal.
  • \sin(90^\circ - \theta) = \cos \theta
  • \tan(90^\circ - \theta) = \cot \theta
  • \sec(90^\circ - \theta) = \csc \theta
• Reciprocal Identities:
  • \csc \theta = \frac{1}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \cot \theta = \frac{1}{\tan \theta}
• Quotient Identities:
  • \tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta}
• Pythagorean Identities:
  • \sin^2 \theta + \cos^2 \theta = 1
  • 1 + \tan^2 \theta = \sec^2 \theta
  • 1 + \cot^2 \theta = \csc^2 \theta

Trigonometric Functions of Real Numbers

• Trigonometric functions can use real numbers as inputs (in radians).

Other Trig Function Properties

• Sine function:
  • Domain: (-\infty, \infty), Range: [-1, 1], Period: 2\pi
• Cosine function:
  • Domain: (-\infty, \infty), Range: [-1, 1], Period: 2\pi
• Even/Odd Functions:
  • Even: cosine
  • Odd: sine, tangent
• Periodic Function: f(t + c) = f(t)