Introduction to Sine Rule for Triangles: In trigonometry, another name for the sine rule for triangles are sine law, sine formula, or laws of sine. Sine rule of triangles are the equation which is the lengths of the sides of an uninformed triangles to the sine of its angle. The sine rule of triangles are as follows, `A/sinalpha=B/sinbeta=C/singamma`, Here A, B, and C are the side lengths of the triangle, and a, ß, and ? are the opposite angles of the triangle. Sometimes trigonometry sine rule of triangles are stated using the reciprocal of this equation, `sinalpha/A=sinbeta/B=singamma/C`. Proving Sine Rule for Triangles: To prove sine rule for triangle, draw the perpendicular to the side of triangles with the distance of x as shown in the figure, Proving Sine Rule From the above diagram, we saw that, sin a =`x/B`, sin ß =`x/A`, => Bsin a = x, A sin ß = x, => Bsin a = A sin ß, => `sinalpha/A = sin beta/ B` ?(1) Proving Sine Rule From the above diagram, we saw that, sin a =`y/C`, sin ? =`y/A`, => Csin a = y, A sin ? = y, => C sin a = A sin ?, => `sinalpha/A=singamma/C`, ?(2) From equation (1) and (2), we get, `sinalpha/A=sinbeta/B=singamma/C`, By taking reciprocal, we get, `A/sinalpha=B/sinbeta=C/singamma`. Example for Sine Rule for Triangles: Example 1: Find the unknown values as shown in the diagram by using sine rule for triangle. Example of Law of Sines Solution:From the diagram C = 12, ß = 60°, and ? = 30°. By using law of sines, `A/sinalpha=B/sinbeta=C/singamma`, => `B/sinbeta=C/singamma`, => B =`(Csinbeta)/singamma`, => B =`(12xxsin60)/sin30`, => B =`(12xx0.866)/0.5`, => B =`10.392/0.5`, => B = 20.784 Similarly, => `A/sinalpha=C/singamma`, => A =`(Csinalpha)/singamma`, => A =`(12xxsinalpha)/sin30`, => A =`(12xxsinalpha)/0.5`, => A = 24 × sin a ? (1) => `A/sinalpha=B/sinbeta`, => A =`(Bsinalpha)/sinbeta`, => A =`(20.784xxsinalpha)/sin60`, => A =`(12xxsinalpha)/0.866`, => A = 24 × sin a ? (2) From (1) and (2), we get, => 24 × sin a = 24 × sin a, => sin a = 1, => a = 90°. Now plug a = 90° in equation (1), we get, (1) => A = 24 × sin 90°, => A = 24 × 1, => A = 24. Know more Law of Sin i found this interesting link Law of Tangents Hence the solutions are, A = 24, B = 20.784, and a = 90°.
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