(6 – 1) Angle and their Measure

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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

Learning target: To convert between decimals and degrees, minutes, seconds forms

To find the arc length of a circle

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

Positive angles: Counterclockwise

Negative angles: Clockwise

Drawing an angle

360

90

150

200

-90

-135

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

From Geometry:

Therefore: using the unit circle r = 1

 = 180

So, one revolution 360 = 2

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

(a) -45 (b)

(a) 90

(b) 270

Finding the arc length & the sector area of a circle

Arc length (s):

is the central angle.

S

r

Area of a sector (A):

(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:

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 – 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.

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

Finding the values of trig functions

Now we have six trig ratios.

Find the exact value of the trig ratios.

Sin is positive when  is in QI.

=

=

=

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

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 – 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

(sin)(csc) = 1

(cos)(sec) = 1

(tan)(cot) = 1

Fundamental Identities:

(1) Reciprocal identities:

(2) Tangent & cotangent identities:

(3) Pythagorean identities:

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 <
y = cotx
• Period: 
• Domain: All real number but
• Range: - < y <
y = cscx

y = cscx

y = sinx

• Period: 
• Domain: All real number but
• Range: -< y  -1
• or 1 y < 
y = secx
• Period: 
• Domain: All real number but
• Range: -< y  -1
• or 1 y < 
(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

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

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

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

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

2

-2

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.

y = 2 cos(2x - ) – 3

a: |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.

y = -3 sin(2x - /2)

a = 3

P = 

/4 x  5/4

No vertical shift

3

0

-3

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.

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

The graph of a tangent function
• Period: 
• Domain: All real number but
• Range: - < y <

interval:

Tendency of y = a tan(x) graph

y = ½ tan(x)

y = 2 tan(x)

y = tan(x)

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
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.

Graph first

a = 1 b = ½

c = /3

P =

Interval:

- /2< (1/2)x + /3 < /2

a = 1

P = 2

Interval:

-5/3 < x < /3

a = 1

P =

Interval:

The graph of a cotangent function
• y = cot(x)
• Period: 
• interval:
• 0 < x < 
• Domain: All real number but
• Range: - < y <
The tendency of y = a cot(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, )
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.
• 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.
• Graph y = cos(x - /2)
• Period: P = 2/|b|
• Interval: 0
• draw the asymptotes
• Draw a parabola between the asymptotes