8.2 Trigonometric Ratios
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1 8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For the ord: You will e le to find the sine, the osine, nd the tngent of n ute ngle. ell Work: Write eh frtion s deiml rounded to the nerest hundredth. 1. 2/ 2. 7/24 Solve eh eqution. 5.8 x x 8.5 ntiiptory Set: y Similrity, right tringle with given ute ngle, suh s, is similr to every other right tringle with tht sme ute ngle mesure. L X Δ ~ ΔLMN ~ ΔXYZ Y Sine the tringles re similr... The rtio of the short leg to the hypotenuse in eh of these tringles will e the sme LN XZ. LM XY The rtio of the long leg to the hypotenuse in eh of these tringles will e the sme MN YZ. LM XY The rtio of the short leg to the long leg in eh of these tringles will e the sme LN XZ. MN YZ These rtios re lled trigonometri rtios. trigonometri rtio is rtio of the lengths of two sides of right tringle. The three si trigonometri rtios re sine, osine, nd tngent, whih re revited s sin, os, nd tn. Trigonometri Rtios Let e right tringle. The sine, the osine, nd the tngent of the ute ngles re defined s follows. M N Z
2 The sine of n ngle is the rtio of the length of the leg opposite the ngle to the length of the hypotenuse. sin sin The osine of n ngle is the rtio of the length of the leg djent to the ngle to the length of the hypotenuse. os djent Leg os djent Leg The tngent of n ngle is the rtio of the length of the leg opposite the ngle to the length of the leg djent to the ngle. tn tn djent Leg djent Leg n often used method of rememering these is the work SOHHTO. Open the ook to pge 541 nd red exmple 1. Exmple: Write eh trigonometri rtio s frtion nd s deiml rounded to the nerest hundredth.. sin J 60/ os J 11/ J tn K 11/ L d. tn J 60 e. sin K f. os K K White ord tivity: dditionl Prtie: Write eh trigonometri rtio s frtion nd s deiml rounded to the nerest hundredth.. sin Y 5/ os Y 12/ tn Y 5/ X 5 d. sin X 12/ e. os X 5/1 0.8 Y 12 Z f. tn X 12/5 2.4
3 The trigonometri rtios of 0, 45, nd 60 n e found using the speil right tringle formuls. Thus they re NEVER written s rounded off deimls Use ny numer you like for. Let = etermine the mesures of the sides of the tringles.,, 2, 2, 6 Sin 45 = os 45 = 1 2 sin 0 = /6 = ½ sin 60 = os 0 = Tn 45 = / = 1 tn 0 = 6 os 60 = /6 = ½ tn 60 = To find the sin, os, nd/or tn of ll other ngles lultor is neessry. efore you n do ny evlution you must mke sure your lultor is in degree mode. For Grphing lultors or lultors whih hve MOE key: push MOE, use the rrow key to move down to Rdin egree, use the rrow key to move right till egree is highlighted, push ENTER or =. For Non-grphing lultors: look for utton leled RG, push repetedly till you see smll in the disply. There re two types of lultors whih n do trigonometri funtions: lger logi, nd non-lger logi. lger Logi Key Strokes Sin os Tn ( egree ) = Non-lger Logi Key Strokes ( egree ) Sin os Tn Open the ook to pge 541 nd red exmple. Exmple: Use your lultor to find eh trigonometri rtio. Round to the nerest hundredth.. sin os tn White ord tivity: Prtie: Use your lultor to find eh trigonometri rtio. Round to the nerest hundredth.. sin os
4 . tn Trigonometri rtios n e used to find missing side on right tringle given n ute ngle nd side. Open the ook to pge 542 nd red exmple 4. Exmple: Find eh length. Round to the nerest hundredth. (o prolem () s n exmple.).. QR. F Q 10.2 ft 15 R 12.9 m P F 9 Use x to represent. sin 6 = QR/ m E x is djent to < QR = 12.9 sin 6 os 9 = 20/F 10.2 is opposite < QR = m F = 20/os 9 Use tn to solve for x. F = m Tn 15 = 10.2/x = 10.2/tn 15 = 8.07 ft. 6 White ord tivity: Prtie: Find eh length. Round to the nerest hundredth... RQ Q. E 17 m 51 R 9.5 in sin 51 = 17/ sin 42 = RQ/9.5 tn 27 = E/1.6 = m RQ = 6.6 in E = 6.9 m 42 P F m E Open the ook to pge 54 nd red exmple 5. Exmple: The Piltushn in Switzerlnd is the world s steepest rilwy. Its steepest setion mkes n ngle of out 25.6 with the horizontl nd rises out 0.9 km. To the nerest hundredth of kilometer, how long is this setion of the rilwy trk? sin 25.6 = 0.9/ = 0.9/sin km = 2.08 km White ord tivity: Prtie: ontrtor is uilding wheel hir rmp for doorwy tht is 1.2 ft ove the ground. The rmp will mke n ngle of 4.8 with the ground. Find the length of the rmp to the nerest hundredth of foot. Sin 4.8 = 1.2/ = 14.4 ft 1.2 ft
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