Concept Category 2. Trigonometry & The Unit Circle
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1 Cncept Categry 2 Trignmetry & The Unit Circle
2 Skill Checklist Use special right triangles t express values f fr the six trig functins Evaluate sine csine and tangent using the unit circle Slve tw-step trignmetric equatins Understand hw the unit circle values fr sine and csine prduce the graphs f sine and csine; Sketch the graphs f sine, csine and tangent using key features; given a graph f a trig functin write the equatin; understand hw the key features f sine and csine functins cnnect t characteristics f peridic phenmena; write a trignmetric functin t mdel peridic phenmena
3 Gal Prblems Find the trig values with calculatr: sin 40 = sin 55 = sin 10 = sin 87 = Find the angles with calculatr: sin x = find angle x cs x = find angle x Slve fr x: cm 53 x cm 12 cm x cm x cm 18 cm 68
4 Frm a pint 80m frm the base f a twer, the angle f elevatin is 28. Hw tall is the twer? x A ladder that is 20 ft is leaning against the side f a building. If the angle frmed between the ladder and grund is 75, hw far will Cach Jarvis have t crawl t get t the frnt dr when he falls ff the ladder (assuming he falls t the base f the ladder)? Find the angle: 22 cm 42 cm θ
5 Space Shuttle: During its apprach t Earth, the space shuttle s glide angle changes. When the shuttle s altitude is abut 15.7 miles, its hrizntal distance t the runway is abut 59 miles. What is its glide angle? Rund yur answer t the nearest tenth. When the space shuttle is 5 miles frm the runway, its glide angle is abut 19. Find the shuttle s altitude at this pint in its descent. Rund yur answer t the nearest tenth. Find the 6 Trig Functins (Ratis) fr each:
6 The Six Basic Trig functins a adjacent Cs c hyptenuse b ppsite Sin c hyptenuse b ppsite Tan a adjacent Sec Csc Ct 1 cs 1 sin 1 tan Sin b Tan Cs a C is always ppsite f the right angle
7 The sides f a right -angled triangle are given special names: The hyptenuse, the ppsite and the adjacent. The hyptenuse is the lngest side and is always ppsite the right angle. The ppsite and adjacent sides refer t anther angle, ther than the 90.
8 Sin Cs Opp Hyp Adj Hyp hyptenuse ppsite Tan Opp Adj adjacent
9 Trig Functins Fr example evaluate sin 40 using sin key Yu shuld get:
10 D Sine these: Functin Try each f these n yur calculatr: sin 55 = sin 10 = sin 87 = 0.999
11 Where t use these trig functins (ratis).
12 Gal Prblem: cm x cm Hw d we slve x???
13 34 15 cm Ask yurself: In relatin t the angle, what pieces d I have? Oppsite and hyptenuse What trig rati uses Oppsite and Hyptenuse? x cm Hw d we slve x??? (15) (15) sin 34 (15)Sin 34 = x 8.39 cm = x x 15 SINE Set up the equatin and slve:
14 Ex2) 12 cm 53 x cm Ask yurself: In relatin t the angle, what pieces d I have? Ask yurself: (12) Tan 53 = x (12) 12 Oppsite and adjacent What trig rati uses Oppsite and adjacent? tangent Set up the equatin and slve: (12)tan 53 = x cm = x
15 18 cm x cm 68 Ask yurself: In relatin t the angle, what pieces d I have? Adjacent and hyptenuse What trig rati uses adjacent and hyptenuse? csine Set up the equatin and slve: (x) Cs 68 = 18 x (x)cs 68 = 18 cs 68 cs 68 X = 18 cs 68 (x) X = cm
16 Ex) Frm a pint 80m frm the base f a twer, the angle f elevatin is 28. Hw tall is the twer? x Using the 28 angle as a reference, we knw pp. and adj. sides. Use pp adj tan tan 28 = x (tan 28 ) = x 80 (.5317) = x x m
17 Ex 2) A ladder that is 20 ft is leaning against the side f a building. If the angle frmed between the ladder and grund is 75, hw far will Cach Jarvis have t crawl t get t the frnt dr when he falls ff the ladder (assuming he falls t the base f the ladder)? building Using the 75 angle as a reference, we knw hyptenuse and adjacent side. adj Use cs cs 75 = hyp x 20 x 20 (cs 75 ) = x 20 (.2588) = x x ft.
18 Ex 3. When the sun is 62 abve the hrizn, a building casts a shadw 18m lng. Hw tall is the building? x shadw Using the 62 angle as a reference, we knw ppsite and adjacent side. pp Use x tan tan 62 = adj (tan 62 ) = x 18 (1.8807) = x x m
19 Inverse Trig Functin t find the Angle Inverse Sine Functin Using sin -1 (inverse sine): If = sin θ then sin -1 (0.7315) = θ angle
20 Mre Examples: 1. sin x = find angle x. x = sin -1 (0.1115) sin = x = cs x = find angle x x = cs -1 (0.8988) cs = x = 26
21 22 cm 42 cm θ This time, yu re lking fr angle ɵ Ask yurself: In relatin t the angle, what pieces d I have? Oppsite and hyptenuse What trig rati uses ppsite and hyptenuse? sine Set up the equatin : Sin θ = 22/42 Use the inverse functin t find an angle Sin -1 (22/42) = θ = θ
22 Example 2 C 2cm Find an angle that has a tangent (rati) f 2/3 Prcess: B 3cm A I want t find an ANGLE I was given the sides (rati) Tangent is pp/adj TAN -1 (2/3) = 34 Angle A
23 1. H 14 cm We have been given the adjacent and hyptenuse s we use COSINE: Cs A = 6 cm A C Cs A = Cs C = h a 14 6 Cs C = C = cs -1 (0.4286) C = 64.6 adjacent hyptenuse
24 2. Find angle x x 3 cm A 8 cm O Tan A = Tan x = Tan x = a 8 3 Given adj and pp need t use tan: Tan A = ppsite adjacent x = tan -1 (2.6667) x = 69.4
25 D it Nw: Slve the right triangle: c =? angle B =? angle A =? B 3 2 c C A
26 C Slutin: 3 2 (hyptenuse) 2 = (leg) 2 + (leg) 2 c 2 = c 2 = c 2 = 13 c = 13 c 3.6 B Pythagrean Therem c A
27 cntinued Then use a calculatr t find the measure f B: Then find A: tan ma = mb 56.3
28 Gal Prblem: Space Shuttle: During its apprach t Earth, the space shuttle s glide angle changes. When the shuttle s altitude is abut 15.7 miles, its hrizntal distance t the runway is abut 59 miles. What is its glide angle? Rund yur answer t the nearest tenth.
29 Slutin: Yu knw ppsite and adjacent sides. Which trig rati (functin) can yu use? Glide = x tan x = distance t runway pp. adj. 59 miles Use crrect rati altitude 15.7 miles tan x = Substitute values Use inverse functin: Tan-1 (15.7/59) 14.9 When the space shuttle s altitude is abut 15.7 miles, the glide angle is abut 14.9.
30 Part b) When the space shuttle is 5 miles frm the runway, its glide angle is abut 19. Find the shuttle s altitude at this pint in its descent. Rund yur answer t the nearest tenth. The shuttle s altitude is abut 1.7 miles. Glide = 19 tan 19 = tan 19 = 5 tan 19 = distance t runway pp. adj. h 5 h 5 5 miles altitude h Use crrect rati Substitute values 5 Islate h by multiplying by h Apprximate using calculatr
31 Types f Angles The angle that yur line f sight makes with a line drawn hrizntally. Angle f Elevatin Line f Sight Angle f Elev atin Hrizntal Line Angle f Depressin Hrizntal Line Angle f Depressin Line f Sight
32 Example f Angle f Depressin At an altitude f 1,000 ft., a ballnist measures the angle f depressin frm the balln t the landing zne. The measure f the angle is 15 degrees. Hw far is the balln frm the landing zne?
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37 N calculatr : Happy Wednesday! 3 1) ct 60 2) sin 45 3) If sin, cs? 5 Calculatr: D yu remember the special triangles and 6 trig ratis? CC1Review : a) Slve Lg ( x 7) Lg ( x 4) 3 b) Slve Lg x Lg ( x 25) c) ( ) ( 1) 3 d) 2 x x Sketch f x Lg x Slve e 11 12e 0
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39 Unit Circle: Nv 3 rd 2017 Reference Angles Radians vs. Degrees Special Triangles & 6 Trig Ratis
40 Circle: 360 degrees Why is a circle 360 degrees rather than 100? It s thught that the ancient peples f Mesptamia (Sumerians, Akkadians, and Babylnians) invented the 360-degree circle t describe their bservatins f the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn) alng with the sun and mn. They nticed that the sun s annual trek acrss the sky tk 360 days.
41 Nt 365 days? Several ancient calendars used 360 days fr a year and astrnmers nticed that the sun, which fllws an ecliptic path ver the year, seems t advance in that path by apprximately ne degree per day
42 Unit Circle: Definitin 1 y -1 Center: (0,0) 1 Radius f the circle : x 1 always -1
43 Measure f an Angle -1 1 The measure f an angle is determined by the amunt f rtatin frm the initial side t the Initial Side 1 terminal side. -1
44 Cterminal Angles Angles Sectin that have 4.1, the Figure same initial 4.4, Cterminal and sides are cterminal. Angles, pg. 248 Angles and are cterminal. 44
45 (0,1) ( 1,0) 0 als 360 (1,0) (0, 1) 270
46 Reference Angle (R.A.) 0 Ө Ө R.A. Ө 1-1 R.A. Ө 1 R.A. = Ө 180 Ө 270 Ө -1 1 Acute angle frmed by the terminal side and the x-axis R.A. = 180 Ө 270 Ө R.A. 1-1 Ө R.A. 1 R.A. = Ө R.A. = 360 Ө -1
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48 Cterminal Angles 1 Angles that share the same initial and terminal sides Example: 30 and 390-1
49 Psitive and Negative Angles When sketching angles, always use an arrw t shw directin. Psitive Angle- rtates cunter-clckwise Negative Angle- rtates clckwise
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51 D Nw: Find cterminal and reference angles fr each C-terminal angles :
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53 Happy Mnday Find the exact value fr (N calculatr) tan 60 ct 60 sin 30 csc 30 : Find a negative and a psitive cterminal angle, and reference angle fr each :
54 Trig Functins in Unit Circle
55 Reminder: Pythagrean Therem Angle
56 On an x-y plane Thus,a b c x y r
57 It s abut a circle and a triangle. r = radius = 1 The chsen angle is always attached t the rigin (0,0)
58 Unit Circle: circle with center at (0, 0) and radius = 1 x 2 2 y 1 (-1,0) (0,1) (0,-1) (1,0) sin cs tan y 1 x 1 y x
59 Since r = 1 always in Unit Circle cs sin x x r 1 y y r 1 x y Thus : ( x, y) (cs,sin )
60 ( x, y) ( xy, ) ( x, y) ( x, y)
61 sin 0 cs 0 sin 30 cs 30 Answers in exact values! tan 0 tan 30 (1,0) 30
62 sin 45 sin 60 cs 45 cs 60 tan 45 tan 60 45
63 sin 90 cs 90 tan 90 (0,1)
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67 Mst cmmnly used angles 150 π 90 / π / 3 3π / π / 6 1 0,1 π 60 / 3 π / 4 45 π / ,0 180 π 0 0 1,0-1 1 Because they are special 7π / π / 4 7π 315 / π / 3 5π 300 / 3 3π 270 / 2-1 0, 1 11π / 6 330
68 (cs30,sin 30 ) in 4 quadrants
69 (cs45,sin 45 ) in 4 quadrants Yu can just use 1 1 (, ) 2 2
70 Yur assignment fr the rest f this week: memrize the Unit Circle, at least Quadrant I Handut :
71 r 1 and ( x, y) cs,sin 3 1, ,0 3 1, , 2 2 π 2 2, , , , /2 0, , , , r, / , , 2 2 1,0 3 1, 2 2
72 Final Prduct! 3 1, ,0 2 2, , , , , , ,0 3 1, , , , , , , 2 2
73 Happy Thursday 11/9/17 Find the exact values : Find the values : Find the angles cs150 cs 25 sin x sin 570 tan 120 sin 2227 tan 2.38 Find the angles : 3 sin x 2 3 cs 2 1 sin x 2 tan 1 Slve x : 2 2x 3x x 5x 2e 3e cs x 3cs x sin x sin x sin x 1 2 tan x 3 0 :
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78 Radian vs. Degree
79 Radian The radian is the standard unit f angular measure. The length f an arc f a unit circle is numerically equal t the measurement in radians f the angle that it subtends; ne radian is just under 57.3 degrees
80 It is anther way t measure angles. Definitin f Radian: Sectin 4.1, Figure 4.5, Illustratin f One radian is the measure f a central angle that Arc Length, pg. 249 intercepts arc s equal in length t the radius r f the circle. s r 80
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83 S think abut this ne If we want t cnvert 140 t radian (1)( ) The rati f the angle t the whle circle The circumference with radius = 1
84 Frmulas: Cnversins Between Degrees and Radians 1. T cnvert degrees t radians, multiply degrees by 2. T cnvert radians t degrees, multiply radians by
85 Use Use 180 Change Change 7 3 t Radians t degrees degree t rads (radians) rads (radians) t degrees
86 Radian Angles, pg. 249 Cmmn Radian Measures
87 45 in 4Quadrants
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