Angles and Their Measure

Circles: Circumference and Area

The circle is a fundamental geometric shape in trigonometry. Understanding its properties is essential for measuring angles and arc lengths.

• Circumference: The distance around a circle. For a circle of radius r, the circumference is given by:

• Area: The space enclosed by a circle. For a circle of radius r, the area is:

• Arc Length: For a sector with central angle (in radians), the arc length is:

• Area of a Sector: For a sector with central angle (in radians):

• Fractional Sectors: If the sector is of the circle, then and

Example: For a circle of radius 2, the area of a quarter circle is .

Angles: Drawing, Coterminal, Complementary, and Supplementary

Angles are measured in degrees or radians and can be drawn in standard position (vertex at the origin, initial side along the positive x-axis).

• 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 because .

Degree and Radian Measure

Angles can be measured in degrees or radians. Radians are the standard unit in mathematics.

• Conversion:

  • Degrees to radians:

  • Radians to degrees:

• Common Angles:

  Degrees	Radians
  0°	$0$
  30°
  45°
  60°
  90°
  180°
  270°
  360°

Example: Convert to radians: radians.

Degrees, Minutes, and Seconds

For greater precision, degrees are subdivided into minutes and seconds.

• 1 degree () = 60 minutes ()

• 1 minute () = 60 seconds ()

• To convert D° M' S'' to decimal degrees:

Example:

Arc Length and Area of a Sector

Given a circle of radius r and a central angle (in radians):

• Arc Length:

• Area of Sector:

Example: For ft and radians: ft ft2

Distance on the Surface of the Earth

Assuming the Earth is a sphere of radius , the distance between two points at the same longitude is the arc length:

• (where is the difference in latitude in radians)

Example: Pittsburgh ( N) and Charlotte ( N): radians miles

Finding the Radius of the Earth: Eratosthenes' Method

Eratosthenes estimated Earth's radius by measuring the angle of the sun's rays at two cities and the distance between them.

• Let be the distance between cities, the angle in radians.

Example: miles, radians miles

The Unit Circle and Trigonometric Functions

The Unit Circle

The unit circle is a circle of radius 1 centered at the origin. It is fundamental for defining trigonometric functions for all angles.

• A point is on the unit circle if

• For an angle , the coordinates correspond to

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

Primary Trigonometric Functions

For a point on the unit circle corresponding to angle :

• (if )

Example: For , ,

Secondary Trigonometric Functions

Example: For , , ,

Evaluating Trigonometric Functions on the Unit Circle

• : Reduce to an equivalent angle in

Applications of Trigonometric Functions

Trigonometric functions model periodic phenomena such as population cycles, sound waves, and tides.

Example: The deer population is modeled by , where is years since 2010. To find the population in 2012 ():

Trigonometric Functions of Angles

Soh-Cah-Toa: Right Triangle Definitions

For a right triangle with angle :

These definitions apply only to right triangles.

Trigonometric Functions for Points Not on the Unit Circle

Given a point (not necessarily on the unit circle), the six trigonometric functions for the angle in standard position are:

Example: For , , so , , etc.

Finding Other Trigonometric Functions Given One

Given one trigonometric function and a quadrant, use the Pythagorean identity and sign conventions to find the others.

Example: If and , then , , etc.

Pythagorean Identities

These identities are derived from the Pythagorean theorem and are fundamental for simplifying trigonometric expressions.

Reference Angles

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

Example: The reference angle for is , then (Quadrant II).

Graphs of Trigonometric Functions

Graphs of Sine and Cosine

The graphs of and are periodic with period .

• Amplitude: The maximum value of in or

• Period: For or , period is

• Phase Shift: For or , phase shift is

Example: has amplitude 3, period , phase shift

Graphs of Tangent and Cotangent

The tangent and cotangent functions have period .

• has vertical asymptotes at

• has vertical asymptotes at

• For , period is , phase shift

Example: has amplitude 2, period , phase shift

Graphs of Secant and Cosecant

Secant and cosecant are the reciprocals of cosine and sine, respectively. Their graphs have vertical asymptotes where the denominator is zero.

To graph , first graph , then plot the reciprocal values and draw asymptotes where .

Transformations of Trigonometric Graphs

• Vertical Stretch: or stretches the graph by

• Horizontal Stretch/Shrink: or changes the period to

• Phase Shift: or shifts the graph horizontally by

• Vertical Shift: or shifts the graph up or down by

Example: has amplitude 4, period , phase shift , and is reflected over the x-axis.

Tangent as a Slope

The tangent of an angle gives the slope of a line making that angle with the positive x-axis.

• Given a point and angle , the equation of the line is:

Example: Through at :

Summary Table: Trigonometric Functions

Additional info: Some examples and explanations were expanded for clarity and completeness. All formulas are provided in LaTeX with escaped backslashes for compatibility.