Cncept Categry 2 Trignmetry & The Unit Circle
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
Gal Prblems Find the trig values with calculatr: sin 40 = sin 55 = sin 10 = sin 87 = Find the angles with calculatr: sin x = 0.1115 find angle x cs x = 0.8988 find angle x Slve fr x: 34 15 cm 53 x cm 12 cm x cm x cm 18 cm 68
Frm a pint 80m frm the base f a twer, the angle f elevatin is 28. Hw tall is the twer? x 80 28 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
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:
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
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.
Sin Cs Opp Hyp Adj Hyp hyptenuse ppsite Tan Opp Adj adjacent
Trig Functins Fr example evaluate sin 40 using sin key Yu shuld get: 0.642787
D Sine these: Functin Try each f these n yur calculatr: sin 55 = 0.819 sin 10 = 0.174 sin 87 = 0.999
Where t use these trig functins (ratis).
Gal Prblem: 34 15 cm x cm Hw d we slve x???
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:
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 15.92 cm = x
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 = 48.05 cm
Ex) Frm a pint 80m frm the base f a twer, the angle f elevatin is 28. Hw tall is the twer? x 80 28 Using the 28 angle as a reference, we knw pp. and adj. sides. Use pp adj tan tan 28 = x 80 80 (tan 28 ) = x 80 (.5317) = x x 42.5 42.5 m
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)? 20 75 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 5.2 5.2 ft.
Ex 3. When the sun is 62 abve the hrizn, a building casts a shadw 18m lng. Hw tall is the building? x 62 18 shadw Using the 62 angle as a reference, we knw ppsite and adjacent side. pp Use x tan tan 62 = adj 18 18 (tan 62 ) = x 18 (1.8807) = x x 33.9 33.9m
Inverse Trig Functin t find the Angle Inverse Sine Functin Using sin -1 (inverse sine): If 0.7315 = sin θ then sin -1 (0.7315) = θ angle
Mre Examples: 1. sin x = 0.1115 find angle x. x = sin -1 (0.1115) sin -1 0.1115 = x = 6.4 2. cs x = 0.8988 find angle x x = cs -1 (0.8988) cs -1 0.8988 = x = 26
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 31.59 = θ
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
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 = 0.4286 C = cs -1 (0.4286) C = 64.6 adjacent hyptenuse
2. Find angle x x 3 cm A 8 cm O Tan A = Tan x = Tan x = 2.6667 a 8 3 Given adj and pp need t use tan: Tan A = ppsite adjacent x = tan -1 (2.6667) x = 69.4
D it Nw: Slve the right triangle: c =? angle B =? angle A =? B 3 2 c C A
C Slutin: 3 2 (hyptenuse) 2 = (leg) 2 + (leg) 2 c 2 = 3 2 + 2 2 c 2 = 9 + 4 c 2 = 13 c = 13 c 3.6 B Pythagrean Therem c A
cntinued Then use a calculatr t find the measure f B: Then find A: tan 33.7 3 1 2 ma = 180-90 - mb 56.3
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.
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 = 15.7 59 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.
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 5. 1.7 h Apprximate using calculatr
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
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?
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) 2 2 2 c) ( ) ( 1) 3 d) 2 x x Sketch f x Lg x Slve e 11 12e 0
Unit Circle: Nv 3 rd 2017 Reference Angles Radians vs. Degrees Special Triangles & 6 Trig Ratis
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.
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
Unit Circle: Definitin 1 y -1 Center: (0,0) 1 Radius f the circle : x 1 always -1
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
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
(0,1) 90 180 ( 1,0) 0 als 360 (1,0) (0, 1) 270
Reference Angle (R.A.) 0 Ө 90 1 1 90 Ө 180-1 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. -1 1 R.A. = 180 Ө 270 Ө 360-1 R.A. 1-1 Ө R.A. 1 R.A. = Ө + 180-1 R.A. = 360 Ө -1
Cterminal Angles 1 Angles that share the same initial and terminal sides. -1 1 Example: 30 and 390-1
Psitive and Negative Angles 120 210 When sketching angles, always use an arrw t shw directin. Psitive Angle- rtates cunter-clckwise Negative Angle- rtates clckwise
D Nw: Find cterminal and reference angles fr each C-terminal angles :
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 : 210 420
Trig Functins in Unit Circle
Reminder: Pythagrean Therem Angle
On an x-y plane 2 2 2 2 2 2 Thus,a b c x y r
It s abut a circle and a triangle. r = radius = 1 The chsen angle is always attached t the rigin (0,0)
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
Since r = 1 always in Unit Circle cs sin x x r 1 y y r 1 x y Thus : ( x, y) (cs,sin )
( x, y) ( xy, ) ( x, y) ( x, y)
sin 0 cs 0 sin 30 cs 30 Answers in exact values! tan 0 tan 30 (1,0) 30
sin 45 sin 60 cs 45 cs 60 tan 45 tan 60 45
sin 90 cs 90 tan 90 (0,1)
Mst cmmnly used angles 150 π 90 / 2 120 2π / 3 3π / 4 135 5π / 6 1 0,1 π 60 / 3 π / 4 45 π / 6 30 1,0 180 π 0 0 1,0-1 1 Because they are special 7π / 6 210 225 5π / 4 7π 315 / 4 240 4π / 3 5π 300 / 3 3π 270 / 2-1 0, 1 11π / 6 330
(cs30,sin 30 ) in 4 quadrants
(cs45,sin 45 ) in 4 quadrants Yu can just use 1 1 (, ) 2 2
Yur assignment fr the rest f this week: memrize the Unit Circle, at least Quadrant I Handut :
r 1 and ( x, y) cs,sin 3 1, 2 2-1 1,0 3 1, 2 2 2 2, 2 2 π 2 2, 2 2 150 210 1 3, 2 2 120 135 225 240 1 3, 2 2 1-1 270 0,1 1 1 1 60 45 30 1/2 0, 1 2 2 3 2 1 3, 2 2 3 2 315 300 1 3, 2 2 2 2 1 1, r, 2 2 2 2 2 330 2 1/2 0 3 1, 2 2 1 2 2, 2 2 1,0 3 1, 2 2
Final Prduct! 3 1, 2 2-1 1,0 2 2, 2 2 180 150 1 3, 2 2 135 120 1 90 0,1 60 1 3, 2 2 45 30 2 2, 2 2 0 3 1, 2 2 1 1,0 3 1, 2 2 210 2 2, 2 2 225 240 1 3, 2 2-1 270 0, 1 300 315 330 1 3, 2 2 3 1, 2 2 2 2, 2 2
Happy Thursday 11/9/17 Find the exact values : Find the values : Find the angles cs150 cs 25 sin x 0.628 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 1 0 10x 5x 2e 3e 1 0 2 2 cs x 3cs x 1 0 2 2sin x sin x 1 0 2 4sin x 1 2 tan x 3 0 :
Radian vs. Degree
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
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
S think abut this ne If we want t cnvert 140 t radian 140 140 7 2(1)( ) 360 180 9 The rati f the angle t the whle circle The circumference with radius = 1
Frmulas: Cnversins Between Degrees and Radians 1. T cnvert degrees t radians, multiply degrees by 2. T cnvert radians t degrees, multiply radians by 180 180 84
Use Use 180 Change Change 7 3 t Radians t degrees degree t rads (radians). 140 7 140 2.443460953 180 180 9 180 140 rads (radians) t degrees 7 180 1260 420 3 3
Radian Angles, pg. 249 Cmmn Radian Measures 30 45 60 90 180 360 86
45 in 4Quadrants