Chapter 4 4-1 Radian and degree

measurement

ObjectivesO Describe AnglesO Use radian measureO Use degree measure and convert

between and radian measure

Angles O As derived form the Greek languageO Trigonometry means “measurement of

triangles “O Initially, trigonometry dealt with

relationships among the sides and angles of triangles and was used in the development of astronomy , navigation and surveying

O Today, the use has expanded to involve rotations, orbits, waves, vibrations, etc.

Angles

O An angle is determined by rotating a ray (half-line) about its endpoint.

Definitions O The initial side of an angle is the

starting position of the rotated ray in the formation of an angle.

O The terminal side of an angle is the position of the ray after the rotation when an angle is formed.

O The vertex of an angle is the endpoint of the ray used in the formation of an angle.

Standard Position

O An angle is in standard position when the angle’s vertex is at the origin of a coordinate system and its initial side coincides with the positive x-axis.

Positive and negative angles

O A positive angle is generated by a counterclockwise rotation; whereas a negative angle is generated by a clockwise rotation.

Coterminal

O If two angles are coterminal, then they have the same initial side and the same terminal side.

Radian Measure

O The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.

O One way to measure angles is in radians.

O To define a radian, you can use a central angle of a circle, one whose vertex is the center of the circle.

Radian Measure

O Arc length = radius when = 1 radian

Radian

How many radians are in a circle?

O *In general, the radian measure of a central angle θ with radius r and arc length s is

O θ = s/r.O We know that the circumference of a

circle is 2πr. If we consider the arc s as being the circumference, we get θ = 2π r/r =2

RadiansO This means that the circle itself

contains an angle of rotation of 2π radians. Since 2π is approximately 6.28, this matches what we found above. There are a little more than 6 radians in a circle. (2π to be exact.)

Therefore: A circle contains 2π radians. A semi-circle contains π radians of rotation. A quarter of a circle (which is a right angle) contains

2 radians of rotation 𝜋

Radians

Coterminal anglesO Two angles are coterminal when they

have the same initial and terminal sides.

O For instance, the angle 0 and 2 are coterminal, as are the angles .

O You can find an angle that is coterminal to a given angle by adding or subtracting 2.

Example#1

O For the positive angle 13 / 6, subtract 2 to obtain a coterminal angle.

Example#2

O For the positive angle 3 / 4, subtract 2 to obtain a coterminal angle.

Example#3

O For the negative angle –2 / 3, add 2 to obtain a coterminal angle.

Student guided practice

O Do problems 25 and 26 in your book page 261

Degree MeasureO Definition: A degree is a unit of angle

measure that is equivalent to the rotation in 1/360th of a circle.

O Because there are 360° in a circle, and we now know that there are also 2π radians in a circle, then 2π = 360°.

O 360° = 2π radians 2π radians= 360° O 180° = π radians π radians = 180° O 1° = radians 1 radian= 180/

Degree MeasureO To convert radians to degrees,

multiply by .O To convert degrees to radians,

multiply byO 180/.

Example O Example: Convert 120° to radians.O Example: Convert -315° to radians

ExampleO Example: Convert to degrees.O Convert 7 to degrees.

Student guided practice

O Do odd problems form 55-65 in your book page 262

Acute and ObtuseO An acute angle has a measure

betweenO 0 andO (or between 0° and 90°.)O An obtuse angle has a measure

between and π (or between 90° and 180°.)

Example O Example: Find the supplement and

complement of

Arc LengthO Because we already know that with

radian measure θ =r /s,O where s is the arc length, then s = r

θ.

ExampleO Example: Find the length of the arc

that subtends a central angle with measure 120° in a circle with radius 5 inches.

Example

O A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240 degrees, as shown in Figure 4.15.

Student Guided practice

O Do problem 93 and 94 in your book page 263

Linear and angular speed

O 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, then the linear speed of the particle is

O Linear speed = arc length/ time= s/t =

Linear and angular speed

O Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is Angular speed = central angle/ time= t.

ExampleO Example: The circular blade on a

saw rotates at 2400 revolutions per minute.

O (a) Find the angular speed in radians per second.

O (b) The blade has a diameter of 16 inches. Find the linear speed of a blade tip.

Area of a sector

HomeworkO Do problems

27,28,45,46,51,52,56,58,79,85O In your book page 261 and 262

Closure O Today we learned about radian and

degree measure O Next class we are going to learn

about the unit circle