Ch 4 Trigonometric Functions

4.1 Radian and Degree Measure

Trigonometry

• (from Greek) the measurement of triangles

• Originally used in the development of astronomy, navigation and surveying

• After development of calculus and physical science in 17th century, used for rotations and vibrations, sound waves, light rays, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles.

AnglesDetermined by rotating a ray around it’s end

Positive and Negative Angles

Coterminal Angles

Definition of Radian:

One radian (rad) is the measure of a central angle that intercepts an arc s in equal length to the radius of the circle.

Radian Measure

r

s

rs 2

radians 24

2revolution

4

1

radians 2

2revolution

2

1

radians 36

2revolution

6

1

radians 612

2revolution

12

1

radians 2revolution 1

Example: Sketching and Finding Coterminal Angles

4

9

Example: Sketching and Finding Coterminal Angles

6

7

Example: Sketching and Finding Coterminal Angles

3

17

Complementary and Supplementary Angles

7

5 (b)

8

3 (a)

of supplement theand complement thefind possible, If

Degree Measure

rad 180

1

rad 2360

180rad 1

rad 180

Conversion

.180

rad by degreesmultiply :radians convert to To

.rad

180by radiansmultiply :degrees convert to To

Practice: Convert to degrees

5

3

3

5

Practice: Convert to radians

12030

320

Arc Length

• A circle has a radius of 27 inches. Find the length of the arc intercepted by a central angle of 160o

Linear and Angular Speed(Constant speed along a circular path.)

• Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, the linear speed of the particle is

t

s

time

length arcSpeedLinear

• If is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is

t

time

angle centralSpeedAngular

Example:

• The second hand of a clock is 8 cm long. Find the linear speed of the tip of this second hand as it passes around the clock face.

Example:

The circular blade on a saw rotates at 2400 revolutions per minute.

• Find the angular speed in radians per second.

• The blade has a radius of 4 inches. Find the linear speed of a blade tip in inches per second.