Right Triangle Trigonometry
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1 Rigt Tringle Trigonometry Trigonometry comes from te Greek trigon (tringle) nd metron (mesure) nd is te study of te reltion between side lengts nd ngles of tringles. Angles A ry is strigt lf line tt stretces indefinitely in some direction from point of origin, or vertex. An ngle is formed by rotting ry bout its vertex from te initil side to te terminl side. 1 Terminl side vertex Angle Initil side Figure 1: Angle between intitil nd terminl side Te most common mesure of n ngle is degrees, symbolized by smll superscript circle:. Te ngle corresponding to one full revolution mesures Figure : Some ngles Te specil 90 ngle is clled rigt ngle nd is often, s te figures indicte, mrked wit little squre rter tn te usul circle rcs. Rigt Tringles Te sum of te interior ngles of ny tringle is 180. A tringle in wic one ngle is 90 is clled rigt tringle. If we know one oter ngle, A, in rigt tringle, 1 Te only word from tis prgrp you ve to remember is te word ngle.
2 MAT104 Rigt Tringle Trigonometry Kåre S. Gjldbæk ten te tird ngle nd te reltions between te sides of te tringle re uniquely determined. Te trigonometric functions determine tese reltions, s functions of te ngle. ypotenuse (yp) osite () A cent () Figure : Rigt tringle Te sides re nmed s in figure : Te sides cent nd osite to te ngle A re clled old on te cent () nd osite () side, respectively. Te lst side is clled te ypotenuse (yp). From tese tree sides, it is possible to form 6 rtios: nd teir reciprocls: yp, yp, yp, yp, Tese rtios, s functions of A, define te trigonometric functions: sine, cosine, tngent nd teir buddies cosecnt, secnt, cotngent. Our focus is on te first tree. We define sin(a) = yp, cos(a) = yp, tn(a) = A common mnemonic is SOHCAHTOA formed by te first letters in te definitions: Sin=Opp/Hyp, Cos=Adj/Hyp, Tn=Opp/Adj To remember te mnemonic, it my be elpful to imgine it s te newest gentrified neigborood in Brooklyn.
3 MAT104 Rigt Tringle Trigonometry Kåre S. Gjldbæk Exmple 1. Find te sin A, cos A, tn A in te tringle below. Solution. Te cent side is 5, te osite nd te ypotenuse 5 + = 4 So we ve A 5 sin A = 4 = 4 4 cos A = 5 4 = tn A = 5 If we know te vlues of te trigonometric functions, we cn use tt to mesure te side lengts of rigt tringles. Tis s mny prcticl pplictions. For instnce, people in te business of climbing ldders would be lost witout trigonometry. Exmple. A 0 foot ldder lens ginst wll. Te foot of te ldder mkes ngle wit te ground. Given te knowledge tt sin = 0.87 cos = 0.50 tn = 1.7, ow fr up te wll does te ldder rec? Solution. Te ldder forms rigt tringle wit te ground nd te wll. Te side in te drwing is te osite side to te ngle nd te ypotenuse is 0. By SOHCAHTOA, we ve so sin = yp = 0 ft, 0 feet = 0 sin = 0 ft 0.87 = 17.4 ft Most clcultors cn ndle tis, but since tese contrptions re forbidden to us, you will be provided wit some vlues (only one of wic is needed).
4 MAT104 Rigt Tringle Trigonometry Kåre S. Gjldbæk Common Angles Usully, te trigonometric function vlues re irrtionl numbers. Computing te vlues cn be nsty ffir best left to clcultors or lgebr tecers wit too muc spre time. Tere re, owever, few common ngles for wic te trigonometric function vlues cn be esily computed using only te Pytgoren teorem. Te 45 ngle Te interior ngle sum is 180, so if tere is rigt ngle nd 45 ngle, ten te lst ngle is lso 45. Tis mens tt te cent nd te Figure 4: 45 ngle osite side ve te sme lengt. Let te lengt of te ypotenuse be. By te Pytgoren teorem, we ten ve + =, or =, so = =. We cn now find te trigonometric vlues of te 45 ngle: sin 45 = yp = cos 45 = yp = tn 45 = = = = = 1 Te 0 nd ngle Wen rigt tringle s 0 ngle, te lst ngle is. Tis rigt tringle is obtined by cutting n equilterl tringle in lf. Using 4
5 MAT104 Rigt Tringle Trigonometry Kåre S. Gjldbæk o 1 Figure 5: ngle te Pytgoren teorem gin nd our tinker, we rrive t sin = sin 0 = 1 cos = 1 cos 0 = tn = tn 0 = 5
6 MAT104 Rigt Tringle Trigonometry Kåre S. Gjldbæk Problems Problem 1. Tink bout it. (i) Wy is te ypotenuse lwys te longest side in rigt tringle? (ii) Wy cn tringle ve t most one rigt ngle? (iii) Wy must rigt tringle ve exctly two cute ngles? (iv) Wy must te vlues of sine nd cosine of ngles in rigt tringle be between 0 nd 1? (v) Let A nd B be te two cute ngles in rigt tringle. Stte formul for te ngle B in terms of A. Problem. Work out te detils of te trigonometric vlues of te ngles 0 nd. Problem. For te rigt tringles below, stte te exct vlues of sin, cos, tn. (i) (ii) (iii) (iv) 1 Problem 4. Find te exct vlues of te missing sides. (i) (ii) b b (iii) (iv) c 45 c Problem 5. A tree csts 0 feet long sdow. Te ngle formed by te ground nd te line between te tip of te sdow nd te tree top is 75. How tll is te tree? (sin 75 = 0.97, cos 75 = 0.6, tn 75 =.7) Problem 6. A 15 foot ldder lens ginst wll. Te foot of te ldder mkes 65 ngle wit te ground. How fr is te bse of te ldder from te wll? (sin 65 = 0.91, cos 65 = 0.4, tn 65 =.14) Problem 7. Te Empire Stte Building mesures 1,454 feet. From were 5 you stnd, te ngle between te ground nd te top is 10. How fr wy from te building re you? (sin 10 = 0.17, cos 10 = 0.98, tn 10 = 0.18) b 6
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