4.1: Angles and Their Measure

Circles: Circumference and Area

The circumference and area of a circle are foundational concepts in trigonometry, providing the basis for understanding angles and arc length.

• Circumference:

• Area:

• For a sector (portion) of a circle with central angle (in radians):

  • Arc length:

  • Sector area:

• For a sector with angle in degrees, convert to radians first.

Example: For of a circle (), and .

Angles: Drawing, Coterminal, Complementary, and Supplementary Angles

• Drawing Angles: Place the vertex at the origin, initial side along the positive x-axis. Rotate counterclockwise for positive angles, clockwise for negative.

• Coterminal Angles: Angles that differ by a multiple of or radians.

• Complementary Angles: Two angles whose measures add to or radians.

• Supplementary Angles: Two angles whose measures add to or radians.

Example: and are coterminal; and are complementary.

Degree and Radian Measure

• Degrees: A full circle is .

• Radians: A full circle is radians.

• Conversion:

  • Degrees to radians:

  • Radians to degrees:

Example: radians; radians .

Degrees, Minutes, and Seconds

• 1 degree () = 60 minutes ()

• 1 minute () = 60 seconds ()

• To convert DMS to decimal degrees:

• To convert decimal degrees to DMS: Integer part is degrees, multiply decimal by 60 for minutes, repeat for seconds.

Example:

Arc Length and Area of a Sector

• Arc length: (with in radians)

• Sector area:

Example: For ft, radians, ft, ft.

Distance on the Surface of the Earth

• Earth is approximated as a sphere of radius (e.g., $3960$ miles).

• Distance between two points at the same longitude: , where is the difference in latitude in radians.

Example: Pittsburgh ( N), Charlotte ( N): radians, miles.

Finding the Radius of the Earth: Eratosthenes' Method

• Distance between Alexandria and Syene: miles.

• Angle measured: radians.

• Radius: miles.

4.2: The Unit Circle

Definition and Equation

• The unit circle is a circle of radius 1 centered at the origin .

• Equation:

Determining if a Point is on the Unit Circle

• Substitute into .

Example: Is on the unit circle? (so, not on the unit circle).

Primary Trigonometric Functions on the Unit Circle

• For an angle in standard position, the point on the unit circle corresponds to:

  • (if )

Example: For (), , .

Secondary Trigonometric Functions

Example: For , , , , , , .

Evaluating Trigonometric Functions on the Unit Circle

• Use known values for common angles (e.g., , etc.).

Example: ; .

Applications of Trigonometric Functions

• Trigonometric functions model periodic phenomena, such as population cycles.

Example: models deer population, years after 2010. For (2012): .

4.3: Trigonometric Functions of Angles

Soh-Cah-Toa: Right Triangle Definitions

Note: These definitions apply only to right triangles.

Trigonometric Functions for Points Not on the Unit Circle

• Given a point , the distance from the origin is .

• Then:

Example: For , , , , etc.

Finding Other Trigonometric Functions Given One

• Use Pythagorean identities and quadrant information to find all six functions.

Example: If and , , , , etc.

Pythagorean Identities

Reference Angles

• The reference angle is the acute angle formed by the terminal side of and the x-axis.

• Quadrant I:

• Quadrant II: or

• Quadrant III: or

• Quadrant IV: or

Example: Reference angle for : ; reference angle is .

4.4: Graphs of Sine and Cosine

Basic Graphs

• and are periodic with period .

• Range:

• Key points for : , , , ,

• Key points for : , , , ,

Amplitude

• For or , the amplitude is .

• The graph stretches vertically by .

Example: has amplitude 3; has amplitude 4 and is reflected over the x-axis.

Period

• For or , the period is .

• Larger compresses the graph horizontally.

Example: has period .

Phase Shift

• For or , the phase shift is .

• Positive shifts left, negative $c$ shifts right.

Example: has phase shift .

4.5: Graphs of Other Trigonometric Functions

Graphs of Tangent and Cotangent

• has period , vertical asymptotes at .

• has period , vertical asymptotes at .

Example: has amplitude 2, period , phase shift .

Shifts with Tangent and Cotangent

• For , period is , phase shift is .

• Vertical stretch by .

Tangent as a Slope

• The slope of a line making angle with the x-axis is .

• Point-slope form:

Example: Through , angle : , so .

Graphs of Secant and Cosecant

• is the reciprocal of ; vertical asymptotes where .

• is the reciprocal of ; vertical asymptotes where .

Summary Table: Trigonometric Functions on the Unit Circle

Additional info: Table values inferred for completeness and clarity.