Angles and Radian Measures | P a g e Province – Mathematics –Southwest TN Community College An angle is positive if generated by counterclockwise rotation. An angle is negative

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Province – Mathematics –Southwest TN Community College

Angles and Radian Measures

A point is a location or position that has no size or dimension. A line extends indefinitely in both directions and contains an infinite amount of points. A plane is a flat, smooth surface that extends indefinitely in all directions, and contains an infinite number of points and lines. A line segment or segment is part of a line and starts and stops at distinct points called endpoints. A ray consists of a point on a line and all points of the line on one side of the point.

An angle is formed by two rays (or lines) that have a common endpoint. One ray is called the initial side and the other the terminal side.

An acute angle: (0°<θ<90°) An obtuse angle: (90°<θ<180°)

A right angle: (θ=90°,

rotation)

A straight angle: (θ=180°,

rotation)

A

A

A B

A B

A

Point A

Line AB AB

Plane

Line Segment AB

AB

Ray AB AB

B

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The clock to the left shows the hour hand (initial side) at twelve and the minute hand (terminal side) at 3. These two hands are rays and they form an angle. The common endpoint of the two rays is called the vertex of the angle.

Angles are often labeled with lowercase letter Greek letters, such

as alpha(α), beta(β), gamma(γ), and theta(θ). An angle is in standard position if

- its vertex is at the origin of a rectangular coordinate system

- its initial side lies along the positive x-axis.

α is in standard position θ is in standard position

α is positive θ is negative

α

B

θ

A C

Initial Side Terminal Side

Vertex

θ

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An angle is positive if generated by counterclockwise rotation. An angle is negative if generated by clockwise rotation. Measuring Angles –

A degree is a unit of measuring angles. It represents

of a

complete rotation about the vertex. (There are 360 degrees (360°) in a complete rotation or circle).

A right angle is an angle that measures 90 degrees (90°). Two lines that intersect to form a right angle are perpendicular lines. A small square in the corner of the angle is used to indicate a right angle.

A straight line has an angle measure of 180 degrees (180°). A vertical line runs up and down, and a horizontal line runs left and right.

Angle Names: Acute – Angle measures less than 90°, but more than 0° Straight – Angle measures 180° Obtuse – Angle measures more than 90°, but less than 180° Right – Angle measures 90° Fractional parts of angles are measured in minutes and seconds.

- One minute, written 1’ is

degree: 1’ =

- One second, written 1” is

degree: 1”=

Example –

90°

180°

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Many calculators are able to change from degree-minute-second notation (D°M’S”) to a decimal and vice a versa. Measuring Angles in Radians – Another way to measure an angle is in radians. Radian Measure – Consider an arc of length s on a circle of

radius r. The measure of the central angle, θ, that intercepts the

arc is

Terminal Side

Initial Side

r r

r

One radian

One radian is the measure of the central angle of a circle that intercepts an arc equal to the lengths to the radius of the circle. (the radius of the circle is r) A central angle is an angle whose vertex is at the center of the circle.

r

r β

r

r γ r

We find the length of an angle in radians by dividing the length of the intercepted arc by the radius.

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Example – Find the radian measure of θ if the arc length is 15 inches and the radius is 6 inches?

Example – Find the radian measure of θ if the arc length is 42 inches and the radius is 12 inches? Relationship between Degrees and Radians – We know a full rotation around a circle has 360° and we know that

the circumference of a circle with radius, r, is 2πr. Thus the radian measure of a central angle is the circumference of the circle divided by the circles radius, r. We use the formula for radian measure to find the radian measure of the 360° angle.

Because one complete rotation measures 360° and 2π radians,

360°= 2π radians

Dividing both sides by 2, we have

180°= π radians

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Conversion between degrees and radians:

Using the basic relationship π radians = 180°, 2 π radians = 360°

- To convert degrees to radians, multiply degrees by

- To convert radians to degrees, multiply radians by

Angles that are fractions of a complete rotation are usually

expressed in radian measure as fractional multiples of π, rather

than decimal approximations. For example θ =

rather than using

the decimal approximation θ 1.57 Example – Convert each angle in degrees to radians.

1. 30°

2. 90°

3. -135°

4. 60°

5. 270°

6. -300°

Example – Convert each angle in radians to degrees.

1.

2.

3.

4.

5.

6.

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Drawing Angles in Standard Position – To become comfortable with radian measure, consider angles in standard position. Each origin is the vertex and each initial side is along the positive x-axis. Think of the terminal side as the side of the angle as revolving around the origin. Example – Drawing angles in standard position; Note one way to do this is to convert to degrees.

1.

2.

3.

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4.

Terminal Side Radian Measure of Angle Degree Measure of Angle

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The graph below shows what is called the unit circle. It contains the degree measurements and radian measurements

Recall that the x-axis is initial side so when moving counterclockwise the angles are positive, and when moving clockwise the angles are negative, so instead of having 330° we would have -30°, instead of 315° we would have -45°, instead of 300° we would have -60°, and so on. Also the radian measures

would change also so instead of

we would have -

, instead of

we would have -

, and so on.

http://en.wikibooks.org/wiki/File:Unit_circle_angles.svg

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Two angles with the same initial sides but possibly different rotations are called coterminal angles. Every angle has infinitely many coterminal angles. Coterminal Angles – Increasing or decreasing the degree of an angle in standard position by an integer multiply of 360° results in a coterminal

angle. Thus an angle of θ° is coterminal with angles of θ°± 360°k, where k is an integer. Increasing or decreasing the radian measure of an angle in

standard position by an integer multiply of 2π results in a

coterminal angle. Thus an angle of θ radians is coterminal with

angles of θ ± 2πk, where k is an integer.

Two coterminal angles for an angle of θ° can be found by adding

360° to θ° and by subtracting 360° from θ°. Example – Assuming the following angles are in standard position. Find a positive angle less than 360° that is coterminal with each of the following.

1. a 420° angle

2. a -120° angle

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Example – Assuming the following angles are in standard

position. Find a positive angle less than 2π that is coterminal with each of the following.

1. a

2. a

To find a positive coterminal angle less than 360° or 2π, it is sometimes necessary to add or subtract more than one multiple

of 360° or 2π

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Example – Assuming the following angles are in standard

position. Find a positive angle less than 360° or 2π that is coterminal with each of the following.

1. a

2. a

3. a

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The Length of a Circular Arc

Let r be the radius of a circle and θ the non-negative radian measure of a central angle of the circle. The length of the arc intercepted

by the central angle is Example – A circle has a radius a 10 inches. Find the length of the arc intercepted by a central angle of 120°. Example – A circle has a radius a 6 inches. Find the length of the arc intercepted by a central angle of 45°. Express arc length in

terms of π. Then round your answer to the nearest hundreds.

r

θ

x=arc length

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Linear and Angular Speed Think of a carousel it contains four circular rows of animals. As the carousel revolves, the animals in the outer row travel a greater distance per unit time than those in the inner rows. By contrast, all animals, regardless of row, complete the same number of revolutions per unit time. All animals in the four rows travel at the same angular speed. Linear and Angular Speed – If a point is in motion on a circle of

radius r through an angle of θ radians in time t, then its linear

speed is

, where s is the arc length given by , and its

angular speed is

.

Example – If the hard drive in a computer rotates at 3600 rotations per minute. Express the angular speed of a hard drive in

radians per minute. (Note: 1 revolution = 2π radians) We can establish a relationship between linear speed and angular

speed, by dividing both sides of the arc length formula, by t

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Thus, linear speed is the product of the radius and the angular speed. Note we can write linear speed in terms of angular speed. Recall

and

and .

Example – A wind machine used to generate electricity has blades that are 10 feet in length. The propeller is rotating around at 4 revolutions per second. Find the linear speed, in feet per second of the tips of the blades.

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Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is

. We can use the following formula for the length of a

circular arc, , to find the length of the intercepted arc.

where 1 is the radius of the circle and t is the radian measure of the central angle.

Thus the length of the intercepted arc is θ, which is also the

radian measure of the central angle. *So in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc.

θ

(1,0) 0

P(x,y)

(1,0) 0

P(x,y)

θ

When θ is positive the point P is

reached by moving counterclockwise along the unit circle from (1,0)

When θ is negative the point P is

reached by moving clockwise along the unit circle from (1,0)

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The Six Trigonometric Functions –

The inputs of these six functions are θ and the outputs involve the

point , on the unit circle corresponding to t and the coordinates of this point. Trig. Functions have names that are words, rather than single

letters such as , , . For example, the sine of θ is the y-

coordinate of the point P on the unit circle.

The real number y depends on the letter θ and thus is a function

of t. θ really means θ , where sine is the name of the

function and θ, a real number, is an input.

Name Abbreviation Name Abbreviation

sine sin cosecant csc

cosine cos secant sec

tangent tan cotangent cot

Definitions of the Trigonometric Functions in Terms of

Any Circle:

If θ is a real number and , is a point on the unit circle

that corresponds to θ, and r is the radius then

,

,

,

,

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Definitions of the Trigonometric Functions in Terms of a Unit Circle:

If θ is a real number and , is a point on the unit circle

that corresponds to θ, where r is the radius which is 1 then

,

,

,

,

Finding Values of the Trigonometric Functions

If θ is a real number equal to the length of the

intercepted arc of an angle that measures θ

radians and (

,√

) is the point on the

unit circle that corresponds to θ. Find the

values of the six trig. functions at t. Don’t forget to rationalize if needed.

a)

b)

c)

d)

(1,0) 0

P(

,√

)

θ

θ

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e)

f)

Example –

Use the figure to the left to determine the

values of the six trig. functions at

.

a)

b)

g)

h)

i)

j)

(1,0) 0

P= ,

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Domain and Range of Sine and Cosine Functions –

Example – Find (

) , (

) ,

Example – Find (

) , (

) ,

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Trigonometric Functions at

(

)

√

(

) √

(

)

√

(

) √

(

) (

)

Even and Odd Trigonometric Functions

We know that a function is even if and odd if

. We can show that the cosine function is even and sine is odd.

By definition, the coordinates of the points P and Q are as follows:

,

, In the above figure the x-coordinates of P and Q are the same,

thus thus the cosine function is even. In the above figure the y-coordinates of P and Q are negatives of

each other, thus thus the sine function is odd

P

Q

(1, 0) 0

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Even and Odd Trigonometric Functions

Only cosine and secant functions are even all other are odd. Example – Find the value of each trigonometric function:

a)

b)

Fundamental Identities Trigonometric identities are equations that are true for all real numbers for which the trigonometric expressions in the equations are defined.

Reciprocal Identities

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Quotient Identities

Example – Given

and

√

, find the value of

each of the four remaining trigonometric functions.

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Pythagorean Identities –

Example – Given

and

, find the value of

using a trigonometric identity.

Example – Given

and

, find the value of

using a trigonometric identity.

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Periodic Functions A function f is periodic if there exists a positive number p such

that for all in the domain of f. The smallest positive number p for which f is periodic is called the period of f. Periodic Properties of the Sine and Cosine Functions

The sine and cosine functions are periodic functions and have

period 2π.

The secant and cosecant functions have period 2π. Periodic Properties of the Tangent and Cotangent Functions

The tangent and cotangent functions are periodic functions and

have period π. Example – Find the value of each trigonometric function.

a)

b)

c)

d)

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Repetitive Behavior of the Sine, Cosine and Tangent Functions

For any integer n and real number ,

Evaluating Trig. Functions with a Calculator

a)

b)

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Angles and Radian Measures | P a g e Province – Mathematics –Southwest TN Community College An angle is positive if generated by counterclockwise rotation. An angle is negative