Trigonometric Functions

Angles: Basic Terminology

Understanding angles is fundamental in trigonometry. This section introduces the basic geometric concepts necessary for angle measurement and classification.

• Line AB: Determined by two distinct points, A and B, and extends infinitely in both directions.

• Line Segment AB: The portion of line AB between points A and B, including both endpoints.

• Ray AB: Starts at point A and passes through B, continuing infinitely beyond B.

• Angle: Formed by two rays with a common endpoint called the vertex. The rays are the sides of the angle.

Angle Measure and Types

Angles are measured by the amount of rotation from the initial side to the terminal side about the vertex.

• Initial Side: The starting position of the ray.

• Terminal Side: The position after rotation.

• Positive Angle: Rotation is counterclockwise.

• Negative Angle: Rotation is clockwise.

Degree Measure

The degree is the most common unit for measuring angles in trigonometry.

• A complete rotation is 360°.

• One degree is of a complete rotation.

Classification of Angles

• Acute Angle:

• Right Angle:

• Obtuse Angle:

• Straight Angle:

Complementary and Supplementary Angles

Angles can be related by their sums:

• Complementary Angles: Two angles whose measures sum to .

• Supplementary Angles: Two angles whose measures sum to .

Examples

• Complement of 40°:

• Supplement of 40°:

• Solving for unknowns:

  • If , then and the angles are and .

  • If , then and the angles are and .

Degrees, Minutes, and Seconds

Angles can be measured in degrees, minutes, and seconds for greater precision.

• 1 minute (1') is of a degree.

• 1 second (1'') is of a minute, or of a degree.

Examples

• Addition:

• Subtraction:

Converting Between Angle Measures

• To Decimal Degrees:

• To Degrees, Minutes, Seconds: (rounded to the nearest second)

Standard Position and Quadrantal Angles

An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• Quadrantal Angles: Angles whose terminal sides lie along the x-axis or y-axis (e.g., , , ).

Coterminal Angles

Coterminal angles share the same initial and terminal sides but may differ by multiples of .

• General Expression: , where is any integer.

• Example: All angles coterminal with are .

Applications: Angular Velocity

Angular velocity problems often involve converting revolutions per unit time to degrees.

• Example: A disk spins at 480 revolutions per minute. In 2 seconds, it completes revolutions. Each revolution is , so the total angle is .

Additional info: These notes cover foundational concepts in trigonometry, including angle measurement, classification, and applications, suitable for college-level study.