Trigonometry - Activity 21 General Triangle Solution: Given three sides.

Transcription

1 Nme: lss: p 43 Mths Helper Plus Resoure Set. opyright 003 rue. Vughn, Tehers hoie Softwre Trigonometry - tivity 1 Generl Tringle Solution: Given three sides. When the three side lengths '', '' nd '' of tringle re known, then the three internl ngles '', '' nd '' n e found. ( See digrm t right. ) These re the formuls used to solve this type of prolem: The sum of the internl ngles equls 180º º The 'sine rule'... sin sin sin The 'osine rule'... ² ² + ² os or ² ² + ² os or ² ² + ² os We will demonstrte the proedure for solving tringle given three sides, using this tringle s n exmple: Use the osine rule to find the lrgest ngle. (Whih is opposite the lrgest side.) + + os os Find the inverse os of -0.5 using sientifi lultor... os 1 ( 0.5) º. Use the sine rule to find one of the remining ngles. sin sin sin sin sin Find the inverse sin of using sientifi lultor... sin 1 ( ) 8.955º 3. Use the sum of internl ngles rule to find the remining ngle. The sum of the internl ngles equls 180º º so 180º (+) 180º (8.955º º) 180º º º The tringle is now solved. This digrm shows ll of the sides nd ngles: º º 8.955º 4 NOTE: There n only e one ngle in tringle tht is otuse (greter thn 90 ). If tringle hs n otuse ngle, then it will e opposite the lrgest side. The reson for finding it first is tht in the next step we will use the sine rule to find the seond ngle. The inverse sin opertion tht we will use n only give us ute ngles (less thn 90 ), so we void possile wrong nswer y first eliminting the only possiility of n otuse ngle.

2 1) Six tringles re shown elow. For eh tringle, find the three interior ngles. Show ll working steps. Use Mths Helper Plus to orret your work fter solving eh tringle. () 3 () 5 () (d) (e) (f) To orret your work... ) Strt Mths Helper Plus nd lod the R - Tringle_sover_SSS.mhp doument. This doument shows the working steps nd digrm for solving tringles when given three side lengths. 3) Disply the tringle solver options ox Doule lik the mouse in the order to the left of the lultions. ( This re is shded in the digrm elow.) The tringle solver options ox will disply its 'Lengths & ngles' t... lik the 'ler' utton to remove the previous tringle, then lik on the '' edit ox. Now type the length for side '' of your tringle. Repet for '' nd ''. lik the 'pply' utton t the ottom of the edit ox. The lulted vlues will disply on the options ox. lik the 'OK' utton to lose the options ox. The lultions nd tringle digrm will e displyed on your sreen. NOTE: If the digrm eomes too ig for your omputer sreen, press the F10 key to mke it smller. To mke the digrm igger, hold down Shift while you press F10.

3 Nme: lss: p 45 Mths Helper Plus Resoure Set. opyright 003 rue. Vughn, Tehers hoie Softwre Trigonometry - tivity Generl Tringle Solution: Given two sides nd the inluded ngle. The ngle etween two sides of tringle is lled the inluded ngle of those sides. If ny two sides nd the inluded ngle re known for tringle, then the remining side length nd the two unknown ngles n e found. In the digrm (right), the two given sides re 3 nd 4. The inluded ngle etween these sides is 30. These re the formuls used to solve this type of prolem: The sum of the internl ngles equls 180º º The 'sine rule'... sin sin sin The 'osine rule'... ² ² + ² os or ² ² + ² os or ² ² + ² os We will demonstrte the proedure for solving tringle given two sides nd the inluded ngle, using the tringle given ove. These re the steps: 1. Use the osine rule to find the unknown side. + os os Tking the positive squre root Use the sine rule to find the smller of the two unknown ngles. ( smller ngle will e opposite smller side. ) sin sin sin sin 3sin Find the inverse sin of using sientifi lultor... sin 1 ( ) º 3. Use the sum of internl ngles rule to find the remining ngle. The sum of the internl ngles equls 180º º so 180º (+) 180º (30º º) 180º º º The tringle is now solved. This digrm shows ll of the sides nd ngles: º º º

4 1) Six tringles re shown elow. For eh tringle, find the three interior ngles using the method desried on the front of this sheet. Show ll working steps. Use Mths Helper Plus to orret your work fter solving eh tringle. 5 () () () (d) 19 (f) (e) To orret your work... ) Strt Mths Helper Plus nd lod the R - Tringle_sover_SS.mhp doument. This doument shows the working steps nd digrm for solving tringles given two sides nd the inluded ngle. 3) Disply the tringle solver options ox Doule lik the mouse in the order to the left of the lultions. ( This re is shded in the digrm elow.) The tringle solver options ox will disply its 'Lengths & ngles' t (See elow)... lik the 'ler' utton to remove the previous tringle, then enter your given ngle nd two sides. You n enter these in three wys, ut the nswer will e orret s long s the given ngle is etween the given sides. ( In this digrm we re entering sides '' nd '', nd the ngle '' whih is etween '' nd ''. ) To enter vlue, lik on its edit ox, then type the vlue. IMPORTNT Do not use the degree opertor for ngles, eg type 35 degrees s just 35, not s 35. lik the 'pply' utton t the ottom of the dilog ox. The lulted vlues will disply on the options ox. lik the 'OK' utton to lose the options ox. The lultions nd tringle digrm will e displyed on your sreen. NOTE: If the digrm eomes too ig for your omputer sreen, press the F10 key to mke it smller. To mke the digrm igger, hold down Shift while you press F10.

5 Nme: lss: p 47 Mths Helper Plus Resoure Set. opyright 003 rue. Vughn, Tehers hoie Softwre Trigonometry - tivity 3 Generl Tringle Solution: Given two sides nd non-inluded ngle. non-inluded ngle is n internl ngle of tringle tht does not lie etween two given sides. If two sides nd non-inluded ngle re given for tringle, then there my e up to two different possile tringles tht hve these mesurements. For exmple, oth tringles in the digrm elow hve sides 7 nd 5, nd non-inluded ngle of se 1: se : º 30º We will demonstrte the proedure for solving tringle given two sides nd non-inluded ngle, using the tringle given ove. These re the steps: 1. Use the sine rule to find the unknown ngle tht is opposite one of the given sides. Sutrt this ngle from 180º to find seond ngle. Test your lulted ngles y ompring them with the given ngle. In this se, euse side > side, ngle must e > ngle. Sine oth vlues of we hve found re greter thn 30º, we n ept them oth. If either vlue were less thn 30º it would men tht this tringle only hd one solution. sin sin sin sin 7sin se 1: Find the inverse sin of using sientifi lultor... 1 sin (0.7) 44.47º Sutrt this ngle from 180º to find seond vlue for ngle : se : ( 180º º ) º. Use the sum of internl ngles rule to find the remining ngle. The sum of the internl ngles equls 180º º so se 1: 180º (+) 180º (30º º) 180º 74.47º º se : 180º (+) 180º (30º º) 180º º 14.47º se 1: se : 3. Use the sine rule to find the remining unknown side. sin sin sin sin sin sin sin sin These tringles show the se 1 nd se solutions... 5sin sin14.47 sin 30 sin 30 se 1: se :

6 1) In step 1 of the tringle solution on the front of this sheet, sientifi lultor is used to find the inverse sine of 0.7. This is ngle, nd 44.47º. Explin why nother vlue of ngle is given y : ( 180º º ) º ) Given two sides nd non-inluded ngle, sometimes there re two possile tringles s in the exmple on the front of this sheet, ut sometimes there my e only 1, or even none t ll. Explin wht will hppen in step 1 of the proedure for eh of these situtions. 3) Three tringles re shown elow. In eh se, two sides nd non-inluded ngle re given. For eh tringle: () If possile, sketh on the digrms to show how seond tringle n e onstruted with the sme three mesurements. Shde the seond tringle ( if there is one ). () Solve the tringle for the unknown side nd ngles using the method on the front of this sheet. Drw lelled digrms of the tringles To orret your work ) Strt Mths Helper Plus nd lod the Tringle_sover_SS.mhp doument. This doument shows the working steps nd digrm for solving tringles when given two sides nd non-inluded ngle. 3) Disply the tringle solver options ox Doule lik the mouse in the order to the left of the lultions. ( This re is shded in the digrm elow.) The tringle solver options ox will disply its 'Lengths & ngles' t... IMPORTNT Do not use the degree opertor for ngles, eg type 35 degrees s just 35, not s 35. lik the 'ler' utton to remove the previous tringle. Enter the sides nd ngles into three of the white edit oxes. lik the 'pply' utton t the ottom of the edit ox. The lulted vlues will disply on the options ox. lik the 'OK' utton to lose the options ox. The lultions nd tringle digrm will e displyed on your sreen. NOTE: If the digrm eomes too ig for your omputer sreen, press the F10 key to mke it smller. To mke the digrm igger, hold down Shift while you press F10.

7 Nme: lss: p 49 Mths Helper Plus Resoure Set. opyright 003 rue. Vughn, Tehers hoie Softwre Trigonometry - tivity 4 Generl Tringle Solution: Given two ngles nd one side. When two internl ngles nd one side of tringle re known, the other two internl ngle nd the unknown sides n e found. These re the formuls used to solve this type of prolem: The sum of the internl ngles equls 180º º The 'sine rule'... sin sin sin We will demonstrte the proedure for solving tringle given two ngles nd one side, using the tringle given ove. These re the steps: 1. Use the sum of internl ngles rule to find the unknown ngle. The sum of the internl ngles equls 180º º so 180º (+) 180º (30º + 70º) 180º 100º 80º. Use the sine rule to find either of the two unknown sides. Finding side : sin sin sin sin 4sin 80 sin Use the sine rule gin to find the remining unknown side. Finding side : sin sin sin sin 4sin 30 sin The tringle is now solved. This digrm shows ll of the sides nd ngles:

8 1) Six tringles re shown elow. In eh se, two ngles nd one side re given. For eh tringle, find the unknown ngle nd sides. Show ll working steps. Use Mths Helper Plus to orret your work fter solving eh tringle. () () () (d) 73 (e) (f) To orret your work... ) Strt Mths Helper Plus nd lod the R - Tringle_sover_S.mhp doument. This doument shows the working steps nd digrm for solving tringles when given two ngles nd one side. 3) Disply the tringle solver options ox Doule lik the mouse in the order to the left of the lultions. ( This re is shded in the digrm elow.) The tringle solver options ox will disply its 'Lengths & ngles' t... IMPORTNT Do not use the degree opertor for ngles, eg type 35 degrees s just 35, not s 35. lik the 'ler' utton to remove the previous tringle, then enter the two known ngles nd side length into the white edit oxes. lik the 'pply' utton t the ottom of the edit ox. The lulted vlues will disply on the options ox. lik the 'OK' utton to lose the options ox. The lultions nd tringle digrm will e displyed on your sreen. NOTE: If the digrm eomes too ig for your omputer sreen, press the F10 key to mke it smller. To mke the digrm igger, hold down Shift while you press F10.

9 Nme: lss: p 51 Mths Helper Plus Resoure Set. opyright 003 rue. Vughn, Tehers hoie Softwre Trigonometry - tivity 5 Generl Tringle Solution: Given three (x,y) points. y (1,3) Plot three points on the (x,y) plne nd join them with lines. If the points re not in the sme stright line, you will hve reted tringle. This digrm shows the tringle reted y the three points (1,3), (-,-) nd (3,-1): The distne formul n e used to find the distne etween two (x,y) points. For exmple, onsider the points (x 1, y 1 ) nd (x, y ). The distne formul gives the distne 'd' etween them s follows: (-,-) x (3,-1 d ( x x + y y 1) ( 1) We n use the distne formul to find the distne etween eh pir of points mking up our tringle. These distnes re the lengths of the three sides. For the exmple ove, the points re: (1,3), (-,-) nd (3,-1), so we lulte the side lengths y tking these points two t time: For the side joining (1,3) nd (-,-) we hve: d ( x x ) + ( y y ( 1) ( 3) 1 ) Similrly, the side joining (1,3) nd (3,-1) is , nd the side joining (-,-) nd (3,-1)is We now know the three side lengths of the tringle, s shown elow: One the three side mesurements re known, then the internl ngles '', '' nd '' n e found using the proedure for tringle when given the three side lengths. In summry, this method is s follows: 1. Use the osine rule to find the lrgest ngle (whih is opposite the lrgest side).. Use the sine rule to find one of the remining ngles. 3. Use the sum of internl ngles rule to find the remining ngle. For our exmple, the finl result is s follows:

10 1) Six tringles re defined y the (x,y) points given elow. Plot eh set of points on the grph grid elow, then find the unknown ngle nd sides. Show ll working steps. Use Mths Helper Plus to orret your work fter solving eh tringle. () (0,) (,5) (5,) () (-,-3) (-,-5) (5,-3) () (1,5) (-5,4) (-,) (d) (1,-1) (-5,-3) (5,-) (e) (-5,3) (-4,-) (-1,1) (f) (5,0) (-1,) (,-1) 5 y x To orret your work ) Strt Mths Helper Plus nd lod the R - Tringle_sover_3_points.mhp doument. This doument shows the working steps nd digrm for solving tringles when given three (x,y) points. 3) Disply the tringle solver options ox Doule lik the mouse in the order to the left of the lultions. ( This re is shded in the digrm elow.) The tringle solver options ox will disply its '3 Points' t... lik the 'ler' utton to remove the previous tringle, then enter the three (x,y) points into the white edit oxes. lik the 'pply' utton t the ottom of the edit ox. The lultions will pper on the text view, nd the digrm on the grph view of Mths Helper Plus. lik the 'OK' utton to lose the options ox nd view the lultions nd digrm. NOTE: If the digrm eomes too ig for your omputer sreen, press the F10 key to mke it smller. To mke the digrm igger, hold down Shift while you press F10.