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(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 convert from degrees to radians and from radians to degrees To find the area of a sector of a circle

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(6 – 1) Angle and their Measure

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6 1 angle and their measure

(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 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

Initial side

Vocabulary: Initial side & terminal side:

Terminal side

Terminal side

Terminal side

Initial side

Initial side


6 1 angle and their measure

Positive angles: Counterclockwise

Negative angles: Clockwise

Drawing an angle

360

90

150

200

-90

-135


6 1 angle and their measure

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

One radian

r


6 1 angle and their measure

From Geometry:

Therefore: using the unit circle r = 1

 = 180

So, one revolution 360 = 2


6 1 angle and their measure

Converting from degrees to radians & from radians to degrees

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

(a) -45(b)


6 1 angle and their measure

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

(c) radians

(d) 3 radians

(a) 90

(b) 270


6 1 angle and their measure

Special angles in degrees & in radians


6 1 angle and their measure

Finding the arc length & the sector area of a circle

Arc length (s):

is the central angle.

S

r

Area of a sector (A):

Important: is in radians.


6 1 angle and their measure

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

Arc length:

Area of the sector:


6 1 angle and their measure

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

  • Convert the degrees to radians

Arc length:

Area of the sector:

2.


6 1 angle and their measure

(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.


6 1 angle and their measure

Unit circle


6 1 angle and their measure

Recall: trig ratio from Geometry

SOH

CAH

TOA


6 1 angle and their measure

Also, Two special triangles

30, 60, 90 triangle

45, 45, 90 triangle

2

1

2

60

1

90

30

45

1

1

1

45

90

1


6 1 angle and their measure

Using the unit circle

r

y

x


6 1 angle and their measure

Finding the values of trig functions

Now we have six trig ratios.


6 1 angle and their measure

Find the exact value of the trig ratios.

Sin is positive when  is in QI.

=

=

=


6 1 angle and their measure

Sin is positive when  is in QII

0


6 1 angle and their measure

Sin is negative when  is in QIII

0


6 1 angle and their measure

Sin is negative when  is in QIV


6 1 angle and their measure

cos 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


6 1 angle and their measure

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(+, -)


6 1 angle and their measure

Find the exact values of the trig ratios.


6 1 angle and their measure

(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

Signs of trig functions in each quadrant


6 1 angle and their measure

(sin)(csc) = 1

(cos)(sec) = 1

(tan)(cot) = 1


6 1 angle and their measure

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

Therefore,


6 1 angle and their measure

Fundamental Identities:

(1) Reciprocal identities:

(2) Tangent & cotangent identities:

(3) Pythagorean identities:


6 1 angle and their measure

Even-Odd Properties


6 1 angle and their measure

Co-functions:


6 1 angle and their measure

Find the period, domain, and range

y = sinx

  • Period: 2

  • Domain: All real numbers

  • Range: -1  y  1


6 1 angle and their measure

y = cosx

  • Period: 2

  • Domain: All real numbers

  • Range: -1  y  1


6 1 angle and their measure

y = tanx

  • Period: 

  • Domain: All real number but

  • Range: - < y <


6 1 angle and their measure

y = cotx

  • Period: 

  • Domain: All real number but

  • Range: - < y <


6 1 angle and their measure

y = cscx

y = cscx

y = sinx

  • Period: 

  • Domain: All real number but

  • Range: -< y  -1

  • or 1 y < 


6 1 angle and their measure

y = secx

  • Period: 

  • Domain: All real number but

  • Range: -< y  -1

  • or 1 y < 


6 1 angle and their measure

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


6 1 angle and their measure

(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


6 1 angle and their measure

Period and amplitude of y = sinx graph


6 1 angle and their measure

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-axis

Set 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 graph


6 1 angle and their measure

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


6 1 angle and their measure

y = 3 sin2x a = |3|

P: 

3

-3


6 1 angle and their measure

y = cos x


6 1 angle and their measure

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-axis

Set 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 graph


6 1 angle and their measure

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


6 1 angle and their measure

2

-2


6 1 angle and their measure

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

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


6 1 angle and their measure

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.


6 1 angle and their measure

y = 2 cos(2x - ) – 3

a: |2|Horizontal shift: /2 x  3/2,

P:  Vertical shift: 3 units downward


6 1 angle and their measure

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.


6 1 angle and their measure

y = -3 sin(2x - /2)

a = 3

P = 

/4 x  5/4

No vertical shift

3

0

-3


6 1 angle and their measure

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.


6 1 angle and their measure

(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


6 1 angle and their measure

The graph of a tangent function

  • Period: 

  • Domain: All real number but

  • Range: - < y <

interval:


6 1 angle and their measure

Tendency of y = a tan(x) graph

y = ½ tan(x)

y = 2 tan(x)

y = tan(x)


6 1 angle and their measure

  • To graph y = atan(bx + c):

  • 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


6 1 angle and their measure

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

y = ½ tan (x + /4)

a = ½ , b = 1,

c = /4

P =

-3/4

/4

Interval:

One half of the interval is the zero point.


6 1 angle and their measure

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

Graphfirst

a = 1 b = ½

c = /3

P =

Interval:

- /2< (1/2)x + /3 < /2


6 1 angle and their measure

a = 1

P = 2

Interval:

-5/3 < x < /3


6 1 angle and their measure

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

a = 1

P =

Interval:


6 1 angle and their measure

The graph of a cotangent function

  • y = cot(x)

  • Period: 

  • interval:

  • 0 < x < 

  • Domain: All real number but

  • Range: - < y <


6 1 angle and their measure

The tendency of y = a cot(x)

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


6 1 angle and their measure

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, )


6 1 angle and their measure

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


6 1 angle and their measure

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, )


6 1 angle and their measure

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


6 1 angle and their measure

I do (ex) Find the period, interval, and asymptotes and sketch the graph.

  • Graph y = sin(2x - )

  • Period: P = 2/|b|

  • Interval: 0 <2x -  < 2

  • draw the asymptotes

  • Draw a parabola between the asymptotes

1

-1


6 1 angle and their measure

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

  • Graph y = cos(x - /2)

  • Period: P = 2/|b|

  • Interval: 0 <x - /2 < 2

  • draw the asymptotes

  • Draw a parabola between the asymptotes


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