(6 – 1) Angle and their Measure Learning target: To convert between decimals and degrees, minutes, seconds forms To find the arc length of a circle To.

(6 – 1) Angle and their MeasureLearning target: To convert between decimals and degrees, minutes, seconds forms

To find the arc length of a circle

To convert from degrees to radians and from radians to degrees

To find the area of a sector of a circle

To find the linear speed of an object traveling in circular motion

Vocabulary: Initial side & terminal side:

Initial side

Terminal s

ide Terminal side

Initial side

Terminal s

ide

Initial side

90360

-90

150

Positive angles: Counterclockwise

Negative angles: Clockwise

200 -135

Drawing an angle

Another unit for an angle: Radian

Definition: An angle that has its vertex at the center of a circle and that intercepts an arc on the circle equal in length to the radius of the circle has a measure of one radian.

r

r

One radian

2C r

The entire circle has 360

From Geometry:

360 2

180 or 180

180

r

r r

r

Therefore: using the unit circle r = 1

= 180

So, one revolution 360 = 2

Degrees Radians Radians Degrees

Degree multiply by Radian multiply by

Converting from degrees to radians & from radians to degrees

180

180

5150 150

180 6radians

18060

3 3

I do: Convert from degrees to radians or from radians to degrees.

(a) -45 (b) 3

2

You do: Convert from degrees to radians or from radians to degrees.

(a) 90

(b) 270

(c) radians

(d) 3 radians

4

Special angles in degrees & in radians

degrees 0 30 45 60 90 180 360

Radians

06

4

3

2

2

Finding the arc length & the sector area of a circle

S

r

s rArc length (s):

is the central angle.

Area of a sector (A): 21

2A r

Important: is in radians.

(ex) A circle has radius 18.2 cm. Find the arc length and the area if the central angle is 3/8.

Arc length:

s r 21

2A r

3(18.2)

8

540621.4

8

s

cm

Area of the sector:

2

2

1 3(18.2)

2 8

3(165.62)

8

195.1

A

cm

You do: (ex) A circle has radius 18.2 cm. Find the arc length and the area if the central angle is 144.

1. Convert the degrees to radians

2. s r

Arc length: Area of the sector:

21

2A r

(6 – 2) Trigonometric functions & Unit circle

Learning target: To find the values of the trigonometric functions using a point on the unit circle

To find the exact values of the trig functions in different quadrants

To find the exact values of special angles

To use a circle to find the trig functions

Vocabulary:

Unit circle is a circle with center at the origin and the radius of one unit.

Unit circle

Recall: trig ratio from Geometry

cosHypotenuse

adjacent bA

c

tanopposite

Aadjacent

Oppositesin

Hypotenuse

aA

c

SOH

CAH

TOA

Also, Two special triangles

30, 60, 90 triangle

45, 45, 90 triangle

30

60

90

45

45 90

3

4

4

6

2

3

1

2

3

1

2

1

121

1

Using the unit circle

x

yr

siny

r

cosx

r

tany

x

Finding the values of trig functionsNow we have six trig ratios.

Oppositesin

ypotenuse

y

h r

cosypotenuse

adjacent x

h r

tanopposite y

adjacent x

1csc

sin

hypotenuse r

opposite y

1sec

cos

hypotenuse r

adjacent x

1cot

tan

adjacent x

opposite y

0sin 0 0

1

y

r

sin 30 sin6

sin 45 sin4

sin 90 sin2

Find the exact value of the trig ratios.

=

=

=

sin 60 sin3

Sin is positive when is in QI.

Sin is positive when is in QII sin

y

r

2sin120 sin

3

3sin135 sin

4

5sin150 sin

6

sin180 sin

0

2

3

2

3

2

2

0

Sin is negative when is in QIII sin

y

r

7

sin 210 sin6

5sin 225 sin

4

4sin 240 sin

3

3sin 270 sin 1

2

Sin is negative when is in QIV

siny

r

5sin 300 sin

3

7sin 315 sin

4

11sin 330 sin

6

sin 360 sin 2 1

cos is positive when is in QI

cosx

r

cos is negative when is in QII

cos30 cos6

3cos135 cos

4

cos is negative when is in QIII

4cos 240 cos

3

cos is positive when is in QIV

11cos330 cos

6

tany

x tan is positive when is

in QI (+, +)

cos is negative when is in QII(-, +)

cos is negative when is in QIII(-, -)

cos is positive when is in QIV(+, -)

4tan 240 tan

3

tan 60 tan3

3tan135 tan

4

11tan 330 tan

6

Find the exact values of the trig ratios.

csc60 csc3

11

csc330 csc6

csc180 csc

sec30 sec6

5

sec 225 sec4

sec90 sec

2

cot 45 cot4

cot 90 cot

2

4

cot 240 cot3

(6 – 3) Properties of trigonometric functions

Learning target: To learn domain & range of the trig functions

To learn period of the trig functions

To learn even-odd-properties

in Q. sin cos tan csc sec cot

I + + + + + +

II + - - + - -

III - - + - - +

IV - + - - + -

Signs of trig functions in each quadrant

coscot

sin

(sin)(csc) = 1

(cos)(sec) = 1

(tan)(cot) = 1

sintan

cos

coscot

sin

The formula of a circle with the center at the origin and the radius 1 is: 2 2 1x y

sin1

y yy

r

cos1

x xx

r

2 2 2 2sin cos 1y x

Therefore, 2 2sin cos 1

2 2tan 1 sec 2 2cot 1 csc

Fundamental Identities:

sintan

cos

coscot

sin

1csc

sin

1

seccos

1

cottan

2 2sin cos 1

(1) Reciprocal identities:

(2) Tangent & cotangent identities:

(3) Pythagorean identities:

Even-Odd Properties

sin( ) sin

cos( ) cos

tan( ) tan

csc( ) csc

sec( ) sec

cot( ) cot

Co-functions:

Find the period, domain, and range

y = sinx

• Period: 2• Domain: All real

numbers

•Range: -1 y 1

y = cosx

• Period: 2• Domain: All real

numbers

•Range: -1 y 1

y = tanx

• Period: • Domain: All real number but

•Range: - < y <

(2 1)

2

nx

y = cotx

• Period: • Domain: All real number but

•Range: - < y <

x n

y = cscx

• Period: • Domain: All real number but

•Range: - < y -1

or 1 y <

x n

y = sinxy = cscx

y = secx

• Period: • Domain: All real number but

Range: - < y -1

or 1 y <

(2 1)

2

nx

Summary for: period, domain, and range of trigonometric functions

Functions Period Domain Range

y = sinx 2 All real #’s -1 y 1

y = cosx 2 All real #’s -1 y 1

y = tanx All real #’s but - < y <

y = cotx All real #’s but - < y <

y = cscx All real #’s but - < y -1

or 1 y < y = secx All real #’s but - < y -1

or 1 y <

(2 1)

2

nx

x n

x n

(2 1)

2

nx

(6 – 4) Graph of sine and cosine functions

Learning target: To graph y = a sin (bx) & y = a cos (bx) functions using transformations

To find amplitude and period of sinusoidal function

To graph sinusoidal functions using key points

To find an equation of sinusoidal graph

Sine function:

Notes: a function is defined as: y = a sin(bx – c) + d

Period :

Amplitude: a

2P

b

2

3

2

22

5

2

Period and amplitude of y = sinx graph

Graphing a sin(bx – c) +d

a: amplitude = |a| is the maximum depth of the graph above half and below half.

bx – c : shifting along x-axisSet 0 bx – c 2 and solve for x to find the starting and ending point of the graph for 1 perid.

d: shifting along y-axis

Period: one cycle of the graph2

Pb

I do (ex) : Find the period, amplitude, and sketch the graph y = 3 sin2x for 2 periods.

Step 1:a = |3|, b = 2, no vertical or horizontal shift

Step 2: Amplitude: |3| Period:

Step 3: divide the period into 4 parts equally.

Step 4: mark one 4 points, and sketch the graph

2P

b

y = 3 sin2x a = |3|

P: 3

-3

2

2

3

2

25

2

y = cos x

Graphing a cos (bx – c) +d

a: amplitude = |a| is the maximum depth of the graph above half and below half.

bx – c : shifting along x-axisSet 0 bx – c 2 and solve for x to find the starting and ending point of the graph for 1 perid.

d: shifting along y-axis

Period: one cycle of the graph2

Pb

We do: Find the period, amplitude, and sketch the graph

y = 2 cos(1/2)x for 1 periods.

Step 1:a = |2|, b = 1/2, no vertical or horizontal shift

Step 2: Amplitude: |2|

Period:

Step 3: divide the period into 4 parts equally.

Step 4: mark the 4 points, and sketch the graph

2P

b

2

-2

You do: Find the period, amplitude, and sketch the graph

y = 3 sin(1/2)x for 1 periods.

I do: Find the period, amplitude, translations, symmetric, and sketch the graph

y = 2 cos(2x - ) - 3 for 1 period.

Step 1:a = |2|, b = 2

Step 2: Amplitude: |2| Period:

Step 3: shift the x-axis 3 units down.

Step 4: put 0 2x – 2 , and solve for x to find the beginning point and the ending point.

Step 5: divide one period into 4 parts equally.

Step 6: mark the 4 points, and sketch the graph.

2P

b

y = 2 cos(2x - ) – 3a: |2| Horizontal shift: /2 x 3/2,P: Vertical shift: 3 units downward

We do: Find the period, amplitude, translations, symmetric, and sketch the graph

y = -3 sin(2x - /2) for 1 period.

Step 1: graph y = 3 sin(2x - /2) first

Step 2:a = |3|, b = 2, no vertical shift

Step 3: Amplitude: |3| Period:

Step 4: put 0 2x – /2 2 , and solve for x to find the beginning point and ending point.

Step 5: divide one period into 4 parts equally.

Step 6: mark the 4 points, and sketch the graph with a dotted line.

Step 7: Start at -3 on the starting x-coordinates.

2P

b

y = -3 sin(2x - /2)

a = 3

P =

/4 x 5/4

No vertical shift

3

-3

0

You do: Find the period, amplitude, translations, symmetric, and sketch the graph

y = 3 cos(/4)x + 2 for 1 period.

Step 1: graph y = 3 cos(/4)x first

Step 2:a = |3|, b = /4

Step 3: Shift 2 units upward

Step 4: Amplitude: |3| Period:

Step 5: Step 5: divide one period into 4 parts equally.

Step 6: mark the 4 points, and sketch the graph with a dotted line.

2P

b

(6 – 5) Graphing tangent, cotangent, cosecant, and secant functions

Learning target: To graph functions of the form y = a tan(bx) + c and y = a cot(bx) + c

To graph functions of the form y = a csc(bx) + c and y = a sec(bx) + c

• Period: • Domain: All real number but

•Range: - < y <

(2 1)

2

nx

interval: 2 2

x

The graph of a tangent function

Tendency of y = a tan(x) graph

y = tan(x)y = 2 tan(x) y = ½ tan(x)

To graph y = a tan(bx + c):

(1)The period is and

(2) The phase shift is

(3) To find vertical asymptotes for the graph:

solve for x that shows the one period

b

c

b

2 2bx c

I do: Find the period and translation, and sketch the graph

y = ½ tan (x + /4)

a = ½ , b = 1,

c = /4

P = b

2 4 2x

3

4 4x

-3/4 /4

Interval:

One half of the interval is the zero point.

We do: Find the period and translation, and sketch the graph

Graph first

1tan

2 3y x

1tan

2 3y x

a = 1 b = ½

c = /3

P = b

Interval:

- /2< (1/2)x + /3 < /2

a = 1

P = 2

Interval:

-5/3 < x < /3

1tan

2 3y x

1tan

2 3y x

You do: Find the period and translation, and sketch the graph

tan( )4

y x

a = 1

P =

Interval:

The graph of a cotangent function

• y = cot(x)

• Period:

• interval:

0 < x < • Domain: All real number but

•Range: - < y <

x n

x n

The tendency of y = a cot(x)cot( )y x1

cot( )2

y x2cot( )y x

As a gets smaller, the graph gets closer to the asymptote.

Graphing cosecant functions

• Period:

• Interval: 0 < x <

• Domain: all real numbers, but x n

• Range: |y| 1 or

y -1 or y 1

(-, -1] [1, )

Step 1: y = cos(x), graph y = sin(x)

Step 2: draw asymptotes x-intercepts

Step 3: draw a parabola between each asymptote with the vertex at y = 1

Graphing secant functions

• Period:

• Interval: /2 < x < 3/2

• Domain: all real numbers, but

• Range: |y| 1 or

y -1 or y 1

(-, -1] [1, )

(2 1)

2

nx

Graphing secant functions

Step 1: graph y = cos(x)

Step 2: draw asymptotes x-intercepts

Step 3: draw a parabola between each asymptote with the vertex at y = 1

I do (ex) Find the period, interval, and asymptotes and sketch the graph. csc(2 )y x

Graph y = sin(2x - )

•Period: P = 2/|b|

• Interval: 0 <2x - < 2

• draw the asymptotes

•Draw a parabola between the asymptotes

1

-1

You do: Find the period, interval, and asymptotes and sketch the graph.

sec2

y x

Graph y = cos(x - /2)

•Period: P = 2/|b|

• Interval: 0 <x - /2 < 2

• draw the asymptotes

•Draw a parabola between the asymptotes