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ContentPlane Trigonometry
Plane TrigonometrySolution of Triangles718 Rightangled triangles are solved by formula 𝑐^{2}=𝑎^{2}+𝑏^{2} 719{ 𝑎=𝑐⋯ Scalene Triangles720Case IThe equation𝑎= 𝑏701 will determine any one of the four quantities 𝐴, 𝐵, 𝑎, 𝑏 when the remaining three are known. The Ambiguous Case721 When, in Case I, two sides and an acute angle opposite to one of them are given, we have,, from the figure,𝑐Then 𝐶 and 180−𝐶 are the vlues of 𝐶 and 𝐶′, by (622). Also 𝑏=𝑐 𝑎^{2}−𝑐^{2}because 𝑏=𝐴𝐷±𝐷𝐶 722 When an angle 𝐵 is to be determined from the equation 𝑏𝑎 𝑏𝑎is a small fraction; the circular measure of 𝐵 may be approximated to by putting Case IIWhen two sides 𝑏, 𝑐 and the included angle 𝐴 are known, the third side 𝑎 is given by the formula 𝑎^{2}=𝑏^{2}+𝑐^{2}−2𝑏𝑐𝐵−𝐶2= 𝑏−𝑐𝑏+𝑐 𝐴2725 Obtained from 𝑏−𝑐𝑏+𝑐= (701), and then applying (670) and (671). 𝐵+𝐶2having been found from the above equation, and 𝐵+𝐶2being equal to 90°− 𝐴2, we have 𝐵= 𝐵+𝐶2+ 𝐵−𝐶2, 𝐶= 𝐵+𝐶2− 𝐵−𝐶2𝐵 and 𝐶 having been determined 𝑎 can be found by Case I. 726 If the logarithms of 𝑏 and 𝑐 are known, the trouble of taking out 𝑏𝑐, and the formula 727 12(𝐵−𝐶)= 𝜃−𝜋4 𝐴2655 728 Or else the subsidiary angle 𝜃= 𝑐𝑏, and the formula 12(𝐵−𝐶)= 𝜃2 𝐴2643 729 𝑎= (𝑏+𝑐)From the figure in 960, by drawing a perpendicular from 𝐵 to 𝐸𝐶 produced. 730 If 𝑎 be required in terms of 𝑏, 𝑐 and 𝐴 alone, and in a form adapted to logarithmic computation, employ the subsidiary angle 𝜃= 4𝑏𝑐(𝑏+𝑐)^{2}and the formula 𝑎=(𝑏+𝑐) Case IIIWhen the three sides are known, the angles may be found without employing logarithms, from the formula 731𝑏^{2}+𝑐^{2}−𝑎^{2}2𝑏𝑐703 732 If logarithms are to be used, take the formula for 𝐴2, 𝐴2, 𝐴2, (704), and (705). Quadrilateral Inscribed in a Circle733𝑎^{2}+𝑏^{2}−𝑐^{2}−𝑑^{2}2(𝑎𝑏+𝑐𝑑)From 𝐴𝐶^{2}=𝑎^{2}+𝑏^{2}−2𝑎𝑏 2𝑄𝑎𝑏+𝑐𝑑613, 733 735 𝑄= =area of 𝐴𝐵𝐶𝐷 and 𝑠= 12(𝑎+𝑏+𝑐+𝑑) Area= 12𝑎𝑏 12𝑐𝑑 (𝑎𝑐+𝑏𝑑)(𝑎𝑑+𝑏𝑐)(𝑎𝑏+𝑐𝑑)702, 733 737 Radius of circumscribed circle = 14𝑄 713, 734, 736 738 if 𝐴𝐷 bizect the side of the triangle 𝐴𝐵𝐶 in 𝐷, 4△𝑏^{2}−𝑐^{2}739 14(𝑏^{2}+𝑐^{2}+2𝑏𝑐 12(𝑏^{2}+𝑐^{2}− 12𝑎^{2}) 742 If 𝐴𝐷 bisect the angle 𝐴 of a triangle 𝐴𝐵𝐶, 𝐵−𝐶2= 𝑏+𝑐𝑏−𝑐 𝐴2743 𝐴𝐷= 2𝑏𝑐𝑏+𝑐 𝐴2744 𝐴𝐷= 𝑏𝑐= 𝑏^{2}745 𝐵𝐷∼𝐶𝐷= 𝑏^{2}−𝑐^{2}𝑎=𝑎
Sources and Referenceshttps://archive.org/details/synopsisofelementaryresultsinpureandappliedmathematicspdfdrive©sideway ID: 210900007 Last Updated: 9/7/2021 Revision: 0 Ref: References
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