Trigonometric Functions. Concept Category 3

Transcription

1 Trignmetric Functins Cncept Categry 3

2 Gals 6 basic trig functins (gemetry) Special triangles Inverse trig functins (t find the angles) Unit Circle: Trig identities a b c 2 2 2

3 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

4 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.

5 I will start psting extra practice fr CC1, CC2, and CC3 nline if yu want t start preparing fr the finals Suggestin: Fcus n passing tw CCs first

6 Sin Cs Opp Hyp Adj Hyp hyptenuse ppsite Tan Opp Adj adjacent

7 Trig Functins Fr example evaluate sin 40 using sin key Yu shuld get:

8 Sme Sine Practice Functin Try each f these n yur calculatr: sin 55 = sin 10 = sin 87 = 0.999

9 Where t use these trig functins (ratis).

10 Gal Prblem: cm x cm Hw d we slve x???

11 34 15 cm x cm Hw d we slve x??? Ask yurself: In relatin t the angle, what pieces d I have? Oppsite and hyptenuse What trig rati uses Oppsite and Hyptenuse? SINE Set up the equatin and slve: (15) (15) sin 34 x 15 (15)Sin 34 = x 8.39 cm = x

12 Ex2) cm x cm Ask yurself: In relatin t the angle, what pieces d I have? Ask yurself: Oppsite and adjacent What trig rati uses Oppsite and adjacent? (12) Tan 53 = x (12) 12 (12)tan 53 = x cm = x tangent Set up the equatin and slve:

13 x cm Ask yurself: In relatin t the angle, what pieces d I have? cm Adjacent and hyptenuse Ask yurself: What trig rati uses adjacent and hyptnuse? csine Set up the equatin and slve: (x) Cs 68 = 18 (x) x (x)cs 68 = 18 cs 68 cs 68 X = 18 cs 68 X = cm

14 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

15 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.

16 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

17 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

18 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

19 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

20 Example 2 C 2cm Find an angle that has a tangent (rati) f 2/3 B 3cm A Prcess: I want t find an ANGLE I was given the sides (rati) Tangent is pp/adj TAN -1 (2/3) = 34 Angle A

21 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

22 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

23 D it Nw: C Slve the right triangle: c =? angle B =? angle A =? B 3 2 c A

24 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

25 cntinued Then use a calculatr t find the measure f B: tan Then find A: ma = mb 56.3

26 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.

27 Slutin: Yu knw ppsite and adjacent sides. Which trig rati (functin) can yu use? Glide = x tan x = pp. distance t runway 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.

28 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 = pp. tan 19 = h distance t runway adj. 5 5 tan 19 = h 5 5 miles altitude h Use crrect rati Substitute values 5 Islate h by multiplying by h Apprximate using calculatr

29 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

30 Nv28 Warm-up: D These Nw Find all key features; sketch : f( x) f( x) x x 2 2 x 1 2x3 2x x2

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33 Unit Circle Intrductin Reminder: Pythagrean Therem Angle

34 On an x-y plane Thus,a b c x y r

35 It s abut a circle and a triangle. r = radius The chsen angle is always attached t the rigin (0,0)

36 Trig Functins + xy crdinate plane Yu need t remember these frmulas fr the final

37 Since x y r if yu think abut it : x y r... Cnics r r r 2 2 x y 1... radius 1 r r Frm yesterday : Thus 2 2 (cs ) (sin ) 1

38 Prf! 2 2 (cs 35 ) (sin 35 )? 2 2 (cs 225 ) (sin 225 )? 2 2 (cs 300 ) (sin 300 )?

39 Hw abut these guys? x y r x x x x y r y y y Rewrite the equatins using Trig functins

40 Prf 1 (tan 30 0 ) 2 (sec30 0 ) 2??? But, yur calculatr desn t have a sec key.

41 1 1 (tan 30 ) ( ) cs

42 Unit Circle (calculatr practice) Try : sin 0 cs 0 sin 45 cs 45 sin 360 cs 360

43 Unit Circle (calculatr practice 2) Try : sin 0.2 cs 0.2 sin cs sin cs 0.707

44 Special Triangles: Find the 6 Trig Functins (Ratis) fr each

45 sin 30 a 1 2a 2 3a 3 cs 30? calculatr 2a 2 a 1 tan 30? calculatr 3a 3 sin 60 cs 60 tan 60 sin 45 cs 45 tan 45 calculatr sin 30? cs 30? tan 30?

46 Nt just fr fractins

47 Unit Circle: circle with center at (0, 0) and radius = x y r r 1 (-1,0) (0,1) (0,-1) (1,0) sin cs tan y r x r y x S pints n this circle must satisfy this equatin.

48 This abut this. cs sin x x r 1 y y r 1 x y Thus : ( x, y) (cs,sin )

49 Handut :

50 Angle first 150 π 90 / π / 3 3π / π / 6 1 0,1 1,0 180 π 0 0 1, π / π / 4 7π / π / 3 5π 300 / 3 3π 270 / 2-1 0, 1 π 60 / 3 π / 4 45 π / π / 6 330

51 ( x, y) ( xy, ) ( x, y) ( x, y)

52 sin 0 sin 30 cs 0 cs 30 tan 0 tan 30 (1,0) 30

53 sin 45 sin 60 cs 45 cs 60 tan 45 tan 60 45

54 sin 90 cs 90 tan 90 (0,1)

55 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

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