MCH T 111 Handout Triangle Review Page 1 of 3


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1 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle: In the tringle shown, ngle is right ngle n sie is the hypotenuse of the tringle. The ommon trigonometri funtions re sine, osine n tngent. They re: os tn sin os tn tn tn os sin os The sum of the interior ngles of tringle is 80, so The other importnt piee of informtion pplile only to right tringles is the Pythgoren Theorem whih gives: + For right tringle, there re 5 quntities tht n e vrie 3 sies n ngles. You must hve two of them in orer to fin the other three. right ngle ontins 90 egrees. Emple: right tringle hs hypotenuse of in. n one of the ngles is 7. Determine the length of eh of the sies of the tringle. 7 y y sin(7 " y " sin(7 5.45" os(7 " " os(7 0.9" hek: (This is suffiient ury for this type of prolem Useful Tringle Reltions: Olique Tringle: tringle in whih none of the ngles is right ngle. There re two tools ville for solving this type of tringle: the lw of sines n the lw of osines. When two ngles n the inlue sie of n olique tringle re known, the other ngle n two sies n e foun using the lw of sines. When two sies n the inlue ngle etween the sies re known, the lw of osines n e use to etermine the thir sie. One the thir sie is known, the lw of sines n e use to fin the other ngles. + os sin sin
2 Hnout Tringle Review Pge of 3 Emple: In the tringle elow, ngle is 3 egrees while ngle is 9 egrees. If sie is in, fin the length of the other two sies n the mgnitue of ngle. in 9 3 NOT TO SLE Fining ngle : Fin sies n using the lw of sines: sin(3 sin sin(9 sin in sin(5 in sin( " sin(5 in sin(9 7.09" sin( Emple: The length of the sies of the tringle elow re 7, 8 n 0 s shown. Fin ngle. + os (8(7 os os (8( Use the lw of sines to fin the other ngles if neee. Trig Funtions Otine from lultors: When solving for ngles in olique tringles, re nees to e tken when using the inverse trig funtions on vlues ner to one. Emple: Fin the ngle α using the lw of sines:.9883 Sin α α 8 40 Mimum Sin α Sin Sin 40 Sin α Most lultors will return inverse sine vlues etween 90 n 90. In this se, Sin  ( α HOWEVER, the orret nswer is relly e reful when working with ngles roun 90 n 80. Minimum 
3 Hnout Tringle Review Pge 3 of 3 Emple: Fin the sine of the two ngles shown in the figure elow Sin (50.70 Sin (30.70 Sin α Mimum.70 Use ution! α Minimum  Sin  ( or 30 Most lultors will give 50 for this lultion! Distne Formul: In the stuy of Sttis, it is often neessry to fin the istne etween two points in spe. This is omplishe y pplying the istne formul to the oorintes of the points. Point (X, Y, Z Y ( X X + ( Y Y + ( Z Z X Z Point (X, Y, Z
4 Hnout Geometry Rules Pge of. Opposite ngles re equl when two stright lines interset. Supplementry ngles totl omplimentry ngles totl stright line interseting two prllel lines proues the following equl ngles 5. The sum of the interior ngles of tringle equls Similr tringles hve the sme shpe D θ D θ 8 θ E E θ 7. irle Equtions Rius, R If D, E 8, n, then y proportion: 8 (8 r Length, S R θ, where θ is in rins irumferene, π D π R re, π D π R 4 Dimeter, D 30 in irle pte from: pplie Mehnis for Engineering Tehnology, Keith M. Wlker, Prentie Hll, 008.
5 Hnout Simultneous Equtions Pge of In the stuy of Sttis, you must e le to solve system of simultneous liner equtions. Emple: For the set of liner equtions given elow, fin the vlues of X n Y tht stisfies eh eqution. 3 X + Y X Y Rtionlize the frtions:.75x +.4Y 0.375X .7Y 4. Tke one of the equtions n solve for X in terms of Y: 3. Sustitute the epression for X into the seon eqution n solve for Y: 4. Sustitute the vlue of Y k into the first eqution n solve for X: 5. Report the nswers:.75x +.4Y 0.75X 0.4Y.375( Y.7Y 4 5.Y.7Y 4.9Y 9 Y 0.75X +.4( Y X.75 X Y.75X X X 8 X 8 Y 0 I generlly work with 4 eiml ples n roun t the en of the prolem!
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