Trignmetric Functins Cncept Categry 3
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
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.
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
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
Sme Sine Practice 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 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
Ex2) 53 12 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 15.92 cm = x tangent Set up the equatin and slve:
x cm Ask yurself: In relatin t the angle, what pieces d I have? 68 18 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 = 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 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
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: C Slve the right triangle: c =? angle B =? angle A =? B 3 2 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: tan 33.7 3 1 2 Then find A: 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 = pp. distance t runway 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 = 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 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
Nv28 Warm-up: D These Nw Find all key features; sketch : f( x) f( x) x x 2 2 x 1 2x3 2x x2
Unit Circle Intrductin 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 The chsen angle is always attached t the rigin (0,0)
Trig Functins + xy crdinate plane Yu need t remember these frmulas fr the final
Since x y r 2 2 2 if yu think abut it : 2 2 2 x y r... Cnics 2 2 2 r r r 2 2 x y 1... radius 1 r r Frm yesterday : Thus 2 2 (cs ) (sin ) 1
Prf! 2 2 (cs 35 ) (sin 35 )? 2 2 (cs 225 ) (sin 225 )? 2 2 (cs 300 ) (sin 300 )?
Hw abut these guys? 2 2 2 x y r x x x 2 2 2 2 2 2 x y r y y y 2 2 2 Rewrite the equatins using Trig functins
Prf 1 (tan 30 0 ) 2 (sec30 0 ) 2??? But, yur calculatr desn t have a sec key.
1 1 (tan 30 ) ( ) cs30 0 2 2 0
Unit Circle (calculatr practice) Try : sin 0 cs 0 sin 45 cs 45 sin 360 cs 360
Unit Circle (calculatr practice 2) Try : sin 0.2 cs 0.2 sin 0.707 cs 0.707 sin 0.707 cs 0.707
Special Triangles: Find the 6 Trig Functins (Ratis) fr each
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?
Nt just fr fractins
Unit Circle: circle with center at (0, 0) and radius = 1 2 2 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.
This abut this. cs sin x x r 1 y y r 1 x y Thus : ( x, y) (cs,sin )
Handut :
Angle first 150 π 90 / 2 120 2π / 3 3π / 4 135 5π / 6 1 0,1 1,0 180 π 0 0 1,0-1 1 7π / 6 210 225 5π / 4 7π / 4 240 4π / 3 5π 300 / 3 3π 270 / 2-1 0, 1 π 60 / 3 π / 4 45 π / 6 315 30 11π / 6 330
( x, y) ( xy, ) ( x, y) ( x, y)
sin 0 sin 30 cs 0 cs 30 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)
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