Unit 1: Angles and Trigonometric Functions

Overview

This unit covers foundational concepts in trigonometry, focusing on angle measurement, relationships between angles, and the evaluation of trigonometric functions. Mastery of these topics is essential for understanding more advanced trigonometric identities and applications.

Angle Measurement and Conversion

Decimal Degrees and Degrees, Minutes, Seconds (DMS)

Angles can be measured in decimal degrees or in degrees, minutes, and seconds (DMS). Understanding how to convert between these formats is crucial for solving trigonometric problems.

• Decimal Degrees: An angle expressed as a decimal (e.g., 45.75°).

• Degrees, Minutes, Seconds (DMS): 1 degree = 60 minutes ('), 1 minute = 60 seconds ("). Example: 45° 45' 0".

• Conversion: To convert from decimal degrees to DMS:

  • Degrees = integer part

  • Minutes = decimal part × 60

  • Seconds = decimal part of minutes × 60

• Example: Convert 36.742° to DMS:

  • Degrees: 36

  • Minutes: 0.742 × 60 = 44.52

  • Seconds: 0.52 × 60 ≈ 31.2

  • Result: 36° 44' 31.2"

Angle Relationships

Complements and Supplements

Angles can be related as complements or supplements, which is useful for solving geometric and trigonometric problems.

• Complementary Angles: Two angles whose measures add up to 90°.

• Supplementary Angles: Two angles whose measures add up to 180°.

• Formulas:

  • Complement:

  • Supplement:

• Example: The complement of 35° is .

Coterminal Angles

Coterminal angles share the same terminal side when drawn in standard position.

• Definition: Angles that differ by integer multiples of 360°.

• Formula: , where is any integer.

• Example: Coterminal angles for 45°: 45°, 405°, -315°, etc.

Angle Relationships

• Reference Angle: The acute angle formed by the terminal side of a given angle and the x-axis.

• Quadrant Location: The sign of trigonometric functions depends on the quadrant in which the terminal side lies.

Trigonometric Functions and Their Values

Trigonometric Functions Defined by a Point

Given a point on the terminal side of an angle , the trigonometric functions can be evaluated as follows (where ):

Example: For the point (3, 4), , so , , .

Trigonometric Functions from Equations

When given an equation involving a trigonometric function, use algebraic manipulation and identities to find other function values.

• Example: If and is in Quadrant II, .

Reciprocal, Pythagorean, and Quotient Identities

These identities are essential tools for relating and finding trigonometric function values.

• Reciprocal Identities:

• Pythagorean Identities:

• Quotient Identities:

Quadrants and Signs of Trigonometric Functions

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:

Example: If is positive and is negative, is in Quadrant II.

Summary Table: Key Trigonometric Identities

Practice and Application

• Be able to define and use all trigonometric terms, rules, and identities.

• Convert between angle measures and identify angle relationships.

• Evaluate trigonometric functions using points, equations, and identities.

• Determine the quadrant of an angle based on function values.

Additional info: Academic context and examples have been added to expand on the brief objectives and ensure the notes are self-contained for exam preparation.