4.1 Angles & Radian Measure

Degrees and Radians

Angles can be measured in degrees or radians. Converting between these units is essential in trigonometry.

• Degrees to Radians: Multiply degrees by .

• Formula:

• Example:

• Radians to Degrees: Multiply radians by .

• Formula:

• Example:

Radian Measure and Arc Length

The radian measure of a central angle is the ratio of the arc length to the radius.

• Formula:

• Example: If , , then radians

Coterminal Angles

Coterminal angles share the same terminal side.

• Formula: or

• Example:

Arc Length

The length of an arc on a circle is proportional to the radius and the central angle (in radians).

• Formula:

• Example: , ,

Angular and Linear Speed

• Angular Speed: (angle per unit time)

• Linear Speed: (distance per unit time along the circle)

4.2 Unit Circle

Coordinates on the Unit Circle

Any point on the unit circle corresponding to angle has coordinates .

• Formula:

Tangent and Reciprocal Functions

• Tangent:

• Reciprocal Functions: , ,

Pythagorean Identities

• Basic Identity:

• Other Identities: ;

4.3 Right Triangle Trigonometry

SOH-CAH-TOA Definitions

• Sine:

• Cosine:

• Tangent:

Finding a Missing Angle

• Inverse Trig Functions: , ,

• Use when the angle is unknown and two sides are known.

4.4 Trig Functions of Any Angle

Distance from Origin

• Formula:

• Used to find the hypotenuse when a point is given.

Six Trig Ratios from a Point

Reference Angles

• Degrees:

  • QII:

  • QIII:

  • QIV:

• Radians:

  • QII:

  • QIII:

  • QIV:

4.5 Sine & Cosine Graphs

General Form of Sine and Cosine

• Formula: or

• Parameters:

  • = amplitude (vertical stretch)

  • = affects period

  • = phase shift (horizontal shift)

  • = vertical shift (midline)

Amplitude

• Formula:

Period

• Formula:

Phase Shift

• Formula:

• Positive: right; Negative: left

Vertical Shift / Midline

• Formula:

• : up; : down

Five Key x-values

• Spacing:

• Divide one period into four equal parts to plot key points.

4.6 Other Trig Graphs

Tangent and Cotangent Graphs

• Tangent Period:

• Phase Shift:

• Asymptotes:

• Cotangent Period:

Secant and Cosecant Graphs

• Secant:

• Cosecant:

• Vertical asymptotes occur where the denominator is zero.

4.7 Inverse Trigonometric Functions

Inverse Sine, Cosine, and Tangent

• Inverse Sine: , principal value

• Inverse Cosine: , principal value

• Inverse Tangent: , principal value

Calculator Mode

• Set calculator to DEG (degrees) or RAD (radians) as required before using trigonometric functions.

4.8 Applications

Solving Right Triangles

• Use SOH-CAH-TOA to find missing sides or angles in right triangles.

• Steps:

  1. Draw the triangle.

  2. Label known and unknown values.

  3. Select the appropriate trigonometric ratio.

Simple Harmonic Motion

• Formula: or

• = amplitude, = angular frequency, = time

Period and Frequency

• Period:

• Frequency:

• Period is the time for one full cycle; frequency is the number of cycles per unit time.