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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

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(6 – 1) Angle and their Measure

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  1. (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

  2. Positive angles: Counterclockwise Negative angles: Clockwise Drawing an angle 360 90 150 200 -90 -135

  3. 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

  4. From Geometry: Therefore: using the unit circle r = 1  = 180 So, one revolution 360 = 2

  5. 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. You do: Convert from degrees to radians or from radians to degrees. (c) radians (d) 3 radians (a) 90 (b) 270

  7. Special angles in degrees & in radians

  8. 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.

  9. (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:

  10. 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.

  11. (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.

  12. Unit circle

  13. Recall: trig ratio from Geometry SOH CAH TOA

  14. 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

  15. Using the unit circle r y x

  16. Finding the values of trig functions Now we have six trig ratios.

  17. Find the exact value of the trig ratios. Sin is positive when  is in QI.  = = =

  18. Sin is positive when  is in QII   0

  19. Sin is negative when  is in QIII  0 

  20. Sin is negative when  is in QIV

  21. 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

  22. 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(+, -)

  23. Find the exact values of the trig ratios.

  24. (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

  25. (sin)(csc) = 1 (cos)(sec) = 1 (tan)(cot) = 1

  26. The formula of a circle with the center at the origin and the radius 1 is: Therefore,

  27. Fundamental Identities: (1) Reciprocal identities: (2) Tangent & cotangent identities: (3) Pythagorean identities:

  28. Even-Odd Properties

  29. Co-functions:

  30. Find the period, domain, and range y = sinx • Period: 2 • Domain: All real numbers • Range: -1  y  1

  31. y = cosx • Period: 2 • Domain: All real numbers • Range: -1  y  1

  32. y = tanx • Period:  • Domain: All real number but • Range: - < y <

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

  34. y = cscx y = cscx y = sinx • Period:  • Domain: All real number but • Range: -< y  -1 • or 1 y < 

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

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

  37. (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

  38. Period and amplitude of y = sinx graph

  39. 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

  40. 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

  41. y = 3 sin2x a = |3| P:  3 -3

  42. y = cos x

  43. 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

  44. 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

  45. 2 -2

  46. You do: Find the period, amplitude, and sketch the graph y = 3 sin(1/2)x for 1 periods.

  47. 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.

  48. y = 2 cos(2x - ) – 3 a: |2| Horizontal shift: /2 x  3/2, P:  Vertical shift: 3 units downward

  49. 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.

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