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(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

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

One radian

r

From Geometry:

Therefore: using the unit circle r = 1

= 180

So, one revolution 360 = 2

Converting from degrees to radians & from radians to degrees

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

(a) -45(b)

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

(c) radians

(d) 3 radians

(a) 90

(b) 270

Special angles in degrees & in radians

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.

(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.

Unit circle

Recall: trig ratio from Geometry

SOH

CAH

TOA

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

Using the unit circle

r

y

x

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.

=

=

=

Sin is positive when is in QII

0

Sin is negative when is in QIII

0

Sin is negative when is in QIV

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(+, -)

Find the exact values of the trig ratios.

(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

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

Therefore,

Fundamental Identities:

(1) Reciprocal identities:

(2) Tangent & cotangent identities:

(3) Pythagorean identities:

Even-Odd Properties

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 <

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 <

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

(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

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-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

y = 3 sin2x a = |3|

P:

3

-3

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-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

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.

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.

(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

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.

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

a = 1

P = 2

Interval:

-5/3 < x < /3

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

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 <x - /2 < 2
- draw the asymptotes
- Draw a parabola between the asymptotes