Section 13.1 Right Triangles

Transcription

1 Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review SOH sin = Reciprocl csc = 2. H cos = Reciprocl sec = 3. TO tn = Reciprocl cot = B. Exmples 1. right tringle hs sides whose lengths re 8-cm, 15-cm, nd 17-cm. Find the vlue of the six trig function of α. 2. Solve: α 12 B T In RST, find m R. S 10 R 4. Find cos θ if sin θ =1/2 5. ssuming the ldder of ldder truck is mounted 8-ft off the ground, wht is the tllest uilding which the 108-ft ldder cn rech using the optimum operting of 60? How fr wy from the uilding would the ldder e plced? 6. flgpole 50-ft high stnds on top of the Lewis Building. From point P in front of rroll s Drugstore, the ngle of elevtion of the top of the pole is nd the ngle of elevtion of the ottom of the pole is How high is the uilding? Honors lger 2 hpter 13 Pge 1

2 . Specil Right Tringles 1. Lel the sides of the tringles with the exct vlue nd then stte the rtio : : : : 2. Does the rtio chnge ecuse the size (length of the sides) of the tringle chnges? 3. Fill in the following chrt with exct vlues for the reference ngles using informtion we discussed tody. Degree Sin os Tn Exmples: Find the exct vlues for the following: ) ) x 12 30º 6 x 60º Homework: p odds, 43, 44, 46, 49, 50 Honors lger 2 hpter 13 Pge 2

3 Section ngles nd ngle Mesure Work Together Drw three different sized circles (see next pge). Drw dimeter through the middle of the circle nd find its length. Find the length of the rdius. Find the circumference of the circle. Mesure the distnce round the circle in terms of the rdius. How mny rdii were there on the first circle? ( leve π in nswer) How mny rdii were there on the second circle? ( leve π in nswer) How mny rdii were there on the third circle? ( leve π in nswer) How mny rdii would go round circle with dimeter of 1500 feet? ( leve π in nswer) Wrp Up If you mesured the distnce round circle in terms of its rdius, wht units of mesure would you give it? In reltionship to circle, if I go hlf wy round the edge of circle how mny of your units of mesure is it? With wht you just lerned, how mny degrees of rottion re there for 3 π of your units of 4 mesure? How mny of your units of mesure re there for degree of rottion of 150º? Other Items Stndrd Position (Initil Sides, Terminl Side nd Vertex) ngle rottion oterminl ngles Find one ngle with positive mesure nd one ngle with negtive mesure coterminl with ech ngle º 7π 2. 3 Reference ngles 1. Find the reference ngles for: Homework p , odds, 26, ll, ll Honors lger 2 hpter 13 Pge 3

4 4 inches 3 inches 2 inches 2 inch rdius 2 inch rdius 2 inch rdius Honors lger 2 hpter 13 Pge 4

5 Section 13.3 Trig Functions of Generl ngles nd Section 13.6 irculr Functions Work Together Use protrctor. Drw n ngle of 30 in the 2 inch rdius circle. Plce the ngle in stndrd position. Lel the point where the terminl ry intersects the circle P(x,y). Find the vlues of x nd y. Wht is sin 30, cos 30, nd tn 30? Repet the steps ove using n ngle of 45 nd rdius of of the new point P? Wht is sin 45, cos 45, nd tn 45? 2 inches. Wht re the coordintes Repet the steps ove using n ngle of 210 nd rdius of 2 inches. Wht re the coordintes of the new point P? Wht is sin 210, cos 210, nd tn 210? n you find the exct vlues? Hint: Think reference tringle! Wrp Up How cn you right sin, cos, nd tn using x, y, nd r (rdius) Sin = os = Updte the following chrt with exct vlues for the reference ngles using informtion we discussed tody nd yesterdy. Rdin Degree Sin os Tn 0 0 π/6 30 π/4 45 π/3 60 π/2 90 Tn = Fill in the following chrt to indicte the sign of trig functions in ech qudrnt. I II III IV sin & csc cos & sec tn & cot S T Honors lger 2 hpter 13 Pge 5

6 Think nd Discuss Drw the ngle in stndrd position, drw the reference ngle, nd find the exct vlue of the trig function. 7π sin 4 tn 240 3π sec 6 csc 270 Find the vlues of the sine nd cosine functions of n ngle in stndrd position with mesure θ if the point with coordintes (3,4) lies on its terminl side. Find sinθ when 5 cosθ = nd the terminl side of θ is in the first qudrnt. 13 If secθ = 2 nd θ lies in qudrnt IV, find tnθ. HW p odds, ll nd p odds, ll Note: when sking for exct vlues this mens do not use your clcultor. Honors lger 2 hpter 13 Pge 6

7 Section 13.4 Lw of Sine Ojectives: 1. To solve non right s (S, S) using the Lw of Sines 2. To solve the miguous non right (SS) using Lw of Sines I. re of Tringle. Formul 1. c 2. B 3. II. Lw of Sines. B. c h B Proof:. Exmples 1. S Solve: XYZ if m X = 32 14, m Y = 57 40, nd z = SS Solve: B if m = 70, = 16, nd = Find the re for the prolems ove. Honors lger 2 hpter 13 Pge 7

8 III. The miguous Tringle (SS). se 1: m < Sitution : If = sin, then 2. Sitution : If < sin, then 3. Sitution c: If > sin, nd, then 4. Sitution d: If > sin, nd < then B. se 2: m Sitution : If, then 2. Sitution : If >, then. Exmples: 1. SS Solve: B, if = 72 14, = 22, nd = SS Solve: B, if = 58, = 14, nd = SS Solve: B, if = 32, = 11, nd = 7. Homework: p.729 3, odds, {29-37 odds, 38 (use the Lw of Sines)}, 40, ll Honors lger 2 hpter 13 Pge 8

9 Section 13.5 Lw of osines Ojectives: 1. To solve non right s (SS nd SSS) using the Lw of osines. Review Lw of osines c B 3. Proof: c h B x - x B. Exmples 1. SS Suppose you wnt to fence tringulr lot s shown. Wht is the length of the fence? SS Solve: XYZ if m X = 39 24, y = 12, nd z = SSS Solve: RST, if r = 19, s = 24.3, nd t = Homework: p odds, 28-30, 33, 36, 37, ll Honors lger 2 hpter 13 Pge 9

10 Section 13.7 Inverse Trig Functions Ojectives: 1. To find vlues of expressions involving inverse trig functions.. Principle Vlues: 1. θ Sin y x θ = ( rcsin y) = Sin y ( x rcsin y) = os y ( rc cos y) = ( x rc cos y) = Tn y ( rc tn y) = ( x rc tn y) x os y θ x Tn y θ = 90 θ 90 = π 2 x π 2 θ = 0 θ 180 = 0 x π θ = 90 < θ < 90 = π 2 < x < π 2 Note: letters re used to the function with domins from the non-restricted trig functions. B. Exmples 1. Write sinθ = r in inverse form. 2. Solve: tn x = rcsin rcsec 3 5. Tn π sin tnrc cos 8 Homework: p.813 1, odds, 46,53, 54, ll Honors lger 2 hpter 13 Pge 10