Section 1.1: Measuring Angles

Basic Terminology

Understanding the terminology of angles is foundational in trigonometry. An angle consists of two rays (or line segments) sharing a common endpoint called the vertex. The rays are called the sides of the angle. The initial side is where the angle starts, and the terminal side is where it ends after rotation.

• Line: Extends infinitely in both directions.

• Line Segment: A part of a line between two endpoints.

• Ray: Starts at an endpoint and extends infinitely in one direction.

• Vertex: The common endpoint of the sides of an angle.

• Initial Side: The starting position of the angle.

• Terminal Side: The position after rotation.

Angles are measured by rotating the initial side to the terminal side. A counterclockwise rotation produces a positive angle, while a clockwise rotation produces a negative angle.

Types of Angles

• Acute Angle:

• Right Angle:

• Obtuse Angle:

• Straight Angle:

Two angles are complementary if their measures sum to , and supplementary if their measures sum to .

Standard Position and Quadrantal Angles

An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis. Quadrantal angles have terminal sides on the x-axis or y-axis (e.g., , , , , ).

Coterminal Angles

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

• To find a coterminal angle:

Degrees, Minutes, and Seconds

Angles can be measured in degrees (), minutes (), and seconds ():

To convert between decimal degrees and degrees-minutes-seconds:

• Decimal to DMS: Multiply decimal part by 60 for minutes, then decimal part of minutes by 60 for seconds.

• DMS to Decimal:

Example: Complement and Supplement

• Complement of :

• Supplement of :

Example: Converting Angle Measures

• Convert to decimal degrees:

• Convert to DMS: , ;

Section 1.2: Angle Relationships and Similar Triangles

Geometric Properties

Understanding relationships between angles and triangles is essential for solving geometric problems in trigonometry.

Vertical Angles

• Vertical angles are formed when two lines intersect; they are always equal in measure.

Angle Pairs with Parallel Lines and a Transversal

Triangles: Types and Properties

• Angle Sum: The sum of the angles in any triangle is .

• Types by Angles:

  • Acute Triangle: All angles are acute.

  • Right Triangle: One angle is .

  • Obtuse Triangle: One angle is obtuse.

• Types by Sides:

  • Equilateral: All sides equal.

  • Isosceles: Two sides equal.

  • Scalene: No sides equal.

Similar and Congruent Triangles

• Similar Triangles: Same shape, not necessarily same size. Conditions:

  • Corresponding angles are equal.

  • Corresponding sides are proportional.

• Congruent Triangles: Same shape and size.

Example: Height Using Similar Triangles

If a tree casts a shadow and a person of known height casts a shadow, set up a proportion to solve for the unknown height:

Section 1.3: Trigonometric Functions on Right Triangles

The Pythagorean Theorem and Distance Formula

• Pythagorean Theorem: In a right triangle, where is the hypotenuse.

• Distance Formula: For points and :

Trigonometric Functions (Coordinate Definition)

For a point on the terminal side of an angle in standard position, with :

Note: always.

Example: Finding Trigonometric Function Values

• Given ,

• , ,

• , ,

Quadrantal Angles and Undefined Values

For quadrantal angles (terminal side on axes), some trigonometric functions are undefined:

• If terminal side is on x-axis: and are undefined.

• If terminal side is on y-axis: and are undefined.

Section 1.4: Trigonometric Identities and Function Properties

Reciprocal Identities

For all angles where the functions are defined:

Signs of Trigonometric Functions by Quadrant

Ranges of Trigonometric Functions

Pythagorean Identities

Quotient Identities

Example: Using Identities

• Given , :

• (since )

Summary Table: Key Trigonometric Concepts

Additional info: Some context and examples were expanded for clarity and completeness, including the full definitions of trigonometric functions, identities, and geometric relationships. All formulas are provided in LaTeX format as required.