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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 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
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.
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. = = =
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
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:
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
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
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.