Trigonometry: Unit Circle Basics

Unit Circle Basics

• The unit circle has key coordinates for angles, which include points represented as (cos x, sin x) for each angle x.
• Important values for trigonometric functions are derived from angles associated with the unit circle.

Common Trigonometric Values

• Cos(π/3) = 1/2
• Cos(2π/3) = -1/2
• Sin(2π/3) = √3/2

Quadrants and Their Characteristics

• Quadrant IV contains positive cosine values.
• Angle 5π/3 is located in Quadrant IV.

Coterminal Angles

• Coterminal angles share the same terminal side. Example: 11π/3 is coterminal with 5π/3.

Reference Points on the Unit Circle

• Focus points on the unit circle are: 0, π/2, 3π/2, and 2π.
• Angles on the unit circle are often expressed in radians.

Trigonometric Identities

• csc(60°) = 1/sin(60°) => csc(60°) = 2√3/3 after rationalizing.
• The relationships among sine, cosine, tangent, cotangent, and cosecant are important in solving trigonometric problems.

Angle Conversion

• 300° is equivalent to 5π/3 radians.
• 120° translates to 2π/3 radians.
• 135° converts to 3π/4 radians.
• 150° corresponds to 5π/6 radians.
• Important angles like 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360° also have specific radian measures.

Differences and Patterns

• The differences in angles for points on the unit circle are typically 15°, aligning common unit circle values for calculations.
• The pattern for the denominators in unit circle values cycles through: 6, 4, 3 in various relationships.

Summary of Key Points

• Knowledge of these angles and their trigonometric values is crucial for solving algebraic problems involving the unit circle.
• True/False examples help confirm understanding of key properties, such as sin(174°) = sin(6°) due to their coterminality.