## This Calculation Equation & Triangle

\( \large A = \sin^{-1} \left[ \frac{a \sin B}{b} \right] \)

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

## Law of Sines Calculator

Uses the law of sines to calculate unknown angles or sides of a triangle. In order to calculate the unknown values you must know and enter 3 known values.

Some calculation choices are redundent but included anyway for exact letter designations.

## Calculation Methods

To 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.

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## Law of Sines

If 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:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

### Equations from Law of Sines solving for angles A, B, and C

\[ A = \sin^{-1} \left[ \frac{a \sin B}{b} \right] \hspace{3em} A = \sin^{-1} \left[ \frac{a \sin C}{c} \right] \]
\[ B = \sin^{-1} \left[ \frac{b \sin A}{a} \right] \hspace{3em} B = \sin^{-1} \left[ \frac{b \sin C}{c} \right] \]
\[ C = \sin^{-1} \left[ \frac{c \sin A}{a} \right] \hspace{3em} C = \sin^{-1} \left[ \frac{c \sin B}{b} \right] \]

### Equations from Law of Sines solving for sides a, b, and c

\[ a = \frac{b \sin A}{\sin B} \hspace{3em} a = \frac{c \sin A}{\sin C} \]
\[ b = \frac{a \sin B}{\sin A} \hspace{3em} b = \frac{c \sin B}{\sin C} \]
\[ c = \frac{a \sin C}{\sin A} \hspace{3em} c = \frac{b \sin C}{\sin B} \]

## Triangle Characteristics

Triangle 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 Reading

Weisstein, Eric W. "Law of Sines" From *MathWorld*-- A Wolfram Web Resource. http://mathworld.wolfram.com/LawofSines.html

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