Introduction to Trigonometry. Algebra 2

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1 Introduction to Trigonometry Algebra 2

2 Angle Rotation Angle formed by the starting and ending positions of a ray that rotates about its endpoint Use θ to represent the angle measure Greek letter theta Counterclockwise rotations give positive θ, angle measures Clockwise rotations give negative θ, angle measures 3c

3 Definitions Standard Position initial side of the angle that lies on the positive x- axis Terminal Side ending position of ray terminates Coterminal Angles Angles that share the same terminal side

4 New Ways to Measure Angles In Geometry, measured angles in degrees In Trigonometry, often measure angles in radians Dealing with angle rotations in trig, which means circles What special number do we associate with circles? Degrees don t contain pi, radians do (more accurate measure)

5 Conversions Convert degrees to radians π 180 Convert radians to degrees 180 π

6 Examples 20 π π π π 6

7 Arc Length Arc Length- part of the circumference Formula s = rθ s is the arc length (how far the point traveled in a circle) r is the radius θ is the measure of the central angle (angle that forms the arc); also known as angular velocity

8 Example A plane is flying in a circular pattern with a radius of 45 miles. How many miles does the plane fly on the circular path over 140 degrees?

9 Example A seat on a Ferris wheel travels one quarter of a revolution. The length of the arc traveled by the seat is 5π feet. Find the radius of the Ferris wheel.

10 Example A point on the Sun s equator makes a full revolution once every days. The Sun has a radius of about 432,200 miles at its equator. What is the angular velocity in radians per hour of a point on the Sun s equator? What distance around the Sun s axis does the point travel in 1 hour?

11 Example A neutron star (an incredibly dense collapsed star) in the Sagittarius Galaxy has a radius of 10 miles and completes a full revolution every seconds. Find the angular velocity of the star in radians per second, then use this velocity to determine how far a point on the equator of the star travels each second.

12 Homework Pg. 833: 7 16

13 Bell Ringer Convert the following from either degrees to radians or vice versa 147 9π 5

14 Basic Trigonometric Functions Sine Cosine Tangent

15 Mnemonics Soh Cah Toa In the order Sine, Cosine, Tangent Oscar Had A Hold On Arthur Oscar Had A Heap Of Apples

16 Special Right Triangles

18 Hand Trick Pinky is the x-axis Thumb is the y-axis Ring Finger is 30 Middle Finger is 45 Pointer Finger is 60

19 Hand Trick Sine is bottom over 2 Cosine is top over 2 Tangent is bottom over top

20 Quadrant I II III IV θ 45 sin θ cos θ tan θ

21 Quadrant I II III IV θ 30 sin θ cos θ tan θ

22 Quadrant I II III IV θ 60 sin θ cos θ tan θ

23 Signs of Each Trig Function in Each Quadrant

24 Examples cos 16π 3 cos 1260 tan 11π 4 sin 600 tan 5π 6 sin 5π 6

25 Homework Pg. 847: 2 9 (together) Worksheet

26 Introduction to Trigonometry: Day 2 Algebra 2

27 Bell Ringer Convert the following from either degrees to radians or vice versa 147 9π 5

28 Basic Trigonometric Functions Sine Cosine Tangent

29 Mnemonics Soh Cah Toa In the order Sine, Cosine, Tangent Oscar Had A Hold On Arthur Oscar Had A Heap Of Apples

30 Special Right Triangles

31 Review Quadrants

32 Who s Positive?

33 Trigonometry Tricks Algebra 2

34 Take a paper plate, one piece of string, and 4 different colored markers or pens

35 Draw an axis on your plate with one of your colors With the same color, write the coordinate points that are on the axis With a new color, write the degree measures With your 3 rd color, write the radian measures With your remaining color, write the letters and quadrant numbers

36 On the Back Write both special triangles

37 Hand Trick Pinky is the x-axis Thumb is the y-axis Ring Finger is 30 Middle Finger is 45 Pointer Finger is 60

38 Hand Trick Sine is bottom over 2 Cosine is top over 2 Tangent is bottom over top

39 Examples cos 16π 3 cos 1260 tan 11π 4 sin 600 tan 5π 6 sin 5π 6

40 Homework Worksheet

41 Graphing Trig Functions Algebra 2

42 Review sin 270 cos 0 cos 930 sin 765 cos 3π tan 630 cos 17π 3 sin 5π 4

43 Trig Graphs Sine Cosine Tangent

44 Definitions Crest/Trough: Max/Min Midline: equation of the line that is halfway between the min and max Amplitude: distance between min and midline, OR distance between max and midline Period: distance it takes to complete one full cycle Frequency: how many cycles it completes in a certain interval Phase Shift: horizontal translations Vertical Shift: vertical translations

45 Standard Form of Sine & Cosine y = a sin bx + c + d y = a cos bx + c + d Amplitude: a Midline: y = d Period: 2π b OR 360 b Phase Shift: c b Vertical Shift: d

46 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 6 cos 4θ

47 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = sin(5θ + 60)

48 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 2 sin 8θ + π 3 + 3

49 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 6 cos θ 3 + 5π 6 2

50 Standard Form of Tangent y = a tan bx + c + d Amplitude: none Midline: y = d Period: π b OR 180 b Phase Shift: c b Vertical Shift: d

51 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 1 tan 2θ

52 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 5 tan 6θ 120 1

53 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 1 6 tan θ 4 + 5π 6 + 1

54 Example Identify the amplitude, midline, period, phase shift, and vertical shift of the following function. y = 8 tan θ 5 + 3π 2 + 1

55 Pythagorean Identity Algebra 2

56 Pythagorean Identities When doing trigonometry, what 2 common shapes are we working with? What is another formula we associate with triangles?

57 Example Given that sin θ = where 0 < θ < π, find cos θ. 2

58 Example Given that cos θ = where π < θ < 3π 2, find sin θ.

59 Example Given that tan θ = where π 2 values of sin θ and cos θ. < θ < π, find the

60 Example Given that tan θ = where 3π 2 values of sin θ and cos θ. < θ < 2π, find the

61 Homework Pg. 858: 1 17

Unit 8 Trigonometry. Math III Mrs. Valentine

Unit 8 Trigonometry. Math III Mrs. Valentine Unit 8 Trigonometry Math III Mrs. Valentine 8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals.