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 + + 80. 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: 5.45 + 0.9 43.97 (This is suffiient ury for this type of prolem Useful Tringle Reltions: 30 3 0 45 45 5 4 3 3 5 7 5 8 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
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 : 80-80 - 3-9 5 Fin sies n using the lw of sines: sin(3 sin sin(9 sin in sin(5 in sin(3 4.475" sin(5 in sin(9 7.09" sin(5 5 4.475 7.09 Emple: The length of the sies of the tringle elow re 7, 8 n 0 s shown. Fin ngle. + os 0 8 + 7 (8(7 os 0 8 7 os (8(7 99.9503 0 7 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 7.805 α 8 40 Mimum Sin α Sin 40 7.805 Sin 40 Sin α.9883 7.805 Most lultors will return inverse sine vlues etween -90 n 90. In this se, Sin - (.9883 8.. 8. 98.78 α HOWEVER, the orret nswer is relly 80-8. 98.7. e reful when working with ngles roun 90 n 80. Minimum -
Hnout Tringle Review Pge 3 of 3 Emple: Fin the sine of the two ngles shown in the figure elow. 50 30 Sin (50.70 Sin (30.70 Sin α Mimum.70 Use ution! 50 30 α Minimum - Sin - (.70 50 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
Hnout Geometry Rules Pge of. Opposite ngles re equl when two stright lines interset. Supplementry ngles totl 80 + 80 3. omplimentry ngles totl 90 + 90 4. stright line interseting two prllel lines proues the following equl ngles 5. The sum of the interior ngles of tringle equls 80 + + 80. 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.
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 0 4 5 3 7 X Y 4 8 0. 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(3.3333.5333Y.7Y 4 5.Y.7Y 4.9Y 9 Y 0.75X +.4(0 0 0.4Y X.75 X 3.3333.5333Y.75X + 4 0.75X X 8 X 8 Y 0 I generlly work with 4 eiml ples n roun t the en of the prolem!