Vocabulary: Initial side & terminal side: Terminal side Terminal side (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

Drawing an angle 360 90 150 200 -90 -135 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 Degrees Radians Radians Degrees Degree multiply by Radian multiply by 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 0 30 45 60 90 180 360 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 (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 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.

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

Sin is negative when  is in QIII 

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.

Learning target: To learn domain & range of the trig functions (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  in Q. sin cos tan csc sec cot I + II - III IV

(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 y = sinx 2 All real #’s -1  y  1 y = cosx y = tanx  All real #’s but - < y < y = cotx y = cscx - < y  -1 or 1 y <  y = secx

To find amplitude and period of sinusoidal function (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 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 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 -3

You do: Find the period, amplitude, translations, symmetric, and 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.

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

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

interval: 0 < x <  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