Section 1.1: Angles

Basic Terminology

Understanding angles and their properties is foundational in trigonometry. Angles are formed by two rays (or line segments) sharing a common endpoint called the vertex.

• Line: Extends infinitely in both directions through two points.

• Line Segment: The portion of a line between two points, including the endpoints.

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

• Angle: Formed by two rays with a common endpoint (vertex).

• Initial Side: The starting position of the ray.

• Terminal Side: The position after rotation.

• Positive Angle: Generated by counterclockwise rotation.

• Negative Angle: Generated by clockwise rotation.

Degree Measure and Types of Angles

• Acute Angle:

• Right Angle:

• Obtuse Angle:

• Straight Angle:

• Complementary Angles: Two angles whose measures sum to .

• Supplementary Angles: Two angles whose measures sum to .

Example: The complement of is . The supplement is .

Degrees, Minutes, and Seconds

• 1 degree () = 60 minutes ()

• 1 minute () = 60 seconds ()

Example:

Converting Between Angle Measures

• To Decimal Degrees:

• To Degrees, Minutes, Seconds: (rounded)

Standard Position and Quadrants

• Standard Position: Vertex at the origin, initial side on the positive x-axis.

• Quadrantal Angles: Terminal side lies on the x- or y-axis (e.g., , , , ).

• Quadrants:

  • QI:

  • QII:

  • QIII:

  • QIV:

Coterminal Angles

Angles that share the same initial and terminal sides but differ by multiples of .

• Formula: , where is any integer.

Example: is coterminal with because .

Application Example: Disk Drive Revolutions

To find the degrees a point moves in 5 seconds at 270 revolutions per minute:

• Degrees per revolution:

• Total revolutions in 5 seconds:

• Total degrees:

Section 1.2: Angle Relationships and Similar Triangles

Geometric Properties

• Vertical Angles: Opposite angles formed by two intersecting lines; always equal.

• Parallel Lines and Transversal: When a transversal crosses parallel lines, several angle pairs are formed:

Triangles

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

• Types of Triangles by Angles:

  • Acute: All angles less than

  • Right: One angle is

  • Obtuse: One angle greater than

• Types by Sides:

  • Equilateral: All sides equal

  • Isosceles: Two sides equal

  • Scalene: No sides equal

Similar and Congruent Triangles

• Similar Triangles: Same shape, corresponding angles equal, sides proportional.

• Congruent Triangles: Same shape and size; all corresponding sides and angles equal.

Conditions for Similarity:

• Corresponding angles are equal.

• Corresponding sides are proportional:

Applications of Similar Triangles

• Finding Unknown Angles: Use the angle sum property and similarity conditions.

• Finding Unknown Sides: Set up proportions using corresponding sides.

• Indirect Measurement: Use similar triangles to find heights or distances (e.g., flagpole height using shadow lengths).

Example: If Joey (63 in. tall) casts a 42 in. shadow and a tree casts a 456 in. shadow, the tree's height is in ft.

Section 1.3: Trigonometric Functions

The Pythagorean Theorem and Distance Formula

• Pythagorean Theorem: In a right triangle with legs , and hypotenuse :

• Distance Formula: For points and :

Definitions of Trigonometric Functions

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

Note: always.

Examples

• Given :

• , , , , ,

Quadrantal Angles

• Angles whose terminal sides lie on the axes (, , , , ).

• Some trigonometric functions are undefined for these angles.

Undefined Functions: If the denominator in the definition is zero, the function is undefined.

Section 1.4: Using the Definitions of the Trigonometric Functions

Reciprocal and Quotient Identities

• Reciprocal Identities:

• Quotient Identities:

Pythagorean Identities

For all angles where the functions are defined:

Signs of Trigonometric Functions by Quadrant

Mnemonic: "All Students Take Calculus" (All positive in I, Sine in II, Tangent in III, Cosine in IV).

Ranges of Trigonometric Functions

Examples Using Identities and Signs

• Given in Quadrant III: , , etc.

• Given in Quadrant II: , ; use identities to find all values.

Determining Possibility of Values

• cot : Possible (cotangent can take any real value).

• cos : Impossible (cosine range is ).

• csc : Impossible (cosecant is undefined for $0$).

Summary Table: Trigonometric Functions and Their Properties

Additional info: Some examples and explanations were expanded for clarity and completeness, including the use of set and interval notation for ranges, and the inclusion of summary tables for quick reference.