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\( A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right]\) A = angle A B = angle B C = angle C a = side a b = side b c = side c P = perimeter s = semi-perimeter K = area r = radius of inscribed circle R = radius of circumscribed circle *Length units are for your reference-only since the value of the resulting lengths will always be the same no matter what the units are. ## Calculator UseUses the law of sines to calculate unknown angles or sides of a triangle. In order to calculate the unknown values you must enter 3 known values. Some calculation choices are redundant but are included anyway for exact letter designations. ## Calculation MethodsTo calculate any angle, A, B or C, say B, enter the opposite side b then another angle-side pair such as A and a or C and c. The performed calculations follow the side side angle (SSA) method and only use the law of sines to complete calculations for other unknowns. To calculate any side, a, b or c, say b, enter the opposite angle B and then another angle-side pair such as A and a or C and c. The performed calculations follow the angle angle side (AAS) method and only use the law of sines to complete calculations for other unknowns.
## Law of SinesIf a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of sines states: \( \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \) ## Equations from Law of Sines solving for angles A, B, and C\( A = \sin^{-1} \left[ \dfrac{a \sin B}{b} \right] \) \( A = \sin^{-1} \left[ \dfrac{a \sin C}{c} \right] \) \( B = \sin^{-1} \left[ \dfrac{b \sin A}{a} \right] \) \( B = \sin^{-1} \left[ \dfrac{b \sin C}{c} \right] \) \( C = \sin^{-1} \left[ \dfrac{c \sin A}{a} \right] \) \( C = \sin^{-1} \left[ \dfrac{c \sin B}{b} \right] \) ## Equations from Law of Sines solving for sides a, b, and c\( a = \dfrac{b \sin A}{\sin B} \) \( a = \dfrac{c \sin A}{\sin C} \) \( b = \dfrac{a \sin B}{\sin A} \) \( b = \dfrac{c \sin B}{\sin C} \) \( c = \dfrac{a \sin C}{\sin A} \) \( c = \dfrac{b \sin C}{\sin B} \) ## Triangle CharacteristicsTriangle perimeter, P = a + b + c Triangle semi-perimeter, s = 0.5 * (a + b + c) Triangle area, K = √[ s*(s-a)*(s-b)*(s-c)] Radius of inscribed circle in the triangle, r = √[ (s-a)*(s-b)*(s-c) / s ] Radius of circumscribed circle around triangle, R = (abc) / (4K) ## References/ Further ReadingWeisstein, Eric W. "Law of Sines" From MathWorld-- A Wolfram Web Resource. Law of Sines. Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 512, 2003. http://hyperphysics.phy-astr.gsu.edu/hbase/lcos.html http://hyperphysics.phy-astr.gsu.edu/hbase/lsin.html |