Chapter 3: Radian Measure and the Unit Circle

3.1 Radian Measure

Understanding radian measure is fundamental to trigonometry and calculus. Radians provide a natural way to measure angles based on the properties of circles.

• Radian: An angle with its vertex at the center of a circle that intercepts an arc equal in length to the radius of the circle has a measure of 1 radian.

• General Formula: If a central angle θ (in radians) intercepts an arc of length s on a circle of radius r, then the radian measure is given by:

• Unit: Radians are dimensionless; they are a ratio of two lengths.

Conversions between Degrees and Radians

• To convert degrees to radians: Multiply the degree measure by .

• To convert radians to degrees: Multiply the radian measure by .

Example: Convert 45° to radians: radians.

Example: Convert radians to degrees: .

Agreement on Angle Measurement Units

• If no unit is specified, angles are assumed to be in radians.

• Be careful: 30° and 30 radians are very different angles.

Equivalent Angle Measures

Common angles and their equivalents in degrees and radians:

Trigonometric Function Values of Angles in Radians

• Trigonometric functions can be evaluated for angles in radians using the unit circle.

• Calculator must be in radian mode when working with radian measures.

3.2 Applications of Radian Measure

Radians are essential for solving real-world problems involving circles, such as arc length and area of a sector.

Arc Length on a Circle

• The length s of an arc intercepted by a central angle θ (in radians) on a circle of radius r is:

• θ must be in radians for this formula to be valid.

Example: Find the arc length for a circle of radius 18.20 cm and central angle 2 radians: cm.

Example: Find the north-south distance between Reno (40° N) and Los Angeles (34° N) on Earth (radius 6400 km): Difference in latitude = 6° = radians. Distance = km.

Area of a Sector of a Circle

• The area A of a sector with radius r and central angle θ (in radians):

• θ must be in radians.

Example: Find the area of a sector with radius 324 m and angle 15°: Convert 15° to radians: radians. m².

Applications: Rope and Gears

• Arc length can be used to determine how much rope is wound around a drum or how far gears rotate.

3.3 The Unit Circle and Circular Functions

The unit circle is a circle of radius 1 centered at the origin. It is fundamental for defining the trigonometric (circular) functions for all real numbers.

Circular Functions

• For a point (x, y) on the unit circle corresponding to an arc length s (or angle θ):

• The equation of the unit circle:

Symmetry and Reference Arcs

• The unit circle is symmetric about the x-axis, y-axis, and the origin.

• Reference arcs help find function values for any angle using known values from the first quadrant.

Function Values as Lengths of Line Segments

• Trigonometric functions can be interpreted as lengths of specific line segments in the unit circle diagram.

3.4 Linear and Angular Speed

Linear and angular speed describe how fast a point moves along a circle and how fast the angle changes, respectively.

Linear Speed

• Linear speed v is the rate at which the arc length s is traversed:

• Alternatively, using angular speed :

Angular Speed

• Angular speed is the rate at which the angle θ changes:

• θ must be in radians, t in units of time.

Applications: Pulley and Satellite

• For a pulley of radius 6 cm rotating at 80 revolutions per minute:

  • Angular speed: radians/minute

  • Linear speed: cm/minute

• For a satellite orbiting Earth at a radius of 8000 km (6400 km + 1600 km):

  • Distance in one orbit:

  • Linear speed: , where T is the period

Reflect: Linear speed measures how fast a point moves along the circumference, while angular speed measures how fast the angle changes at the center.