Law of Cosines Calculator

Law of Cosines Calculator

γ
a
b
c
 
109.47°
Gamma( γ )
5
Side A
6
Side B
9
Side C

What Is Law of Cosines Calculator ?

Uses the law of cosines to calculate unknown sides of a triangle. In order to calculate the unknown values you must enter 3 known values.

Law of Cosines Formula / Calculator

Gamma( γ ) = ?
a = 10
b = 9
c = 3

$$\gamma=cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)$$

$$\gamma=cos^{-1}\left(\frac{10^2+9^2-3^2}{2\ast a\ast b}\right)$$

$$\gamma\approx17.50^{\mathrm o}$$

Gamma( γ ) = 3
a = ?
b = 2
c = 3

$$a=b\ast\cos\gamma\;+\;\sqrt{c^{2\;}\;-\;\;b^2\;\;\ast\;\;\left(\sin\gamma\right)^2}$$

$$a=2\;\ast\;\cos\left(3^\circ\right)\;+\;\sqrt{3^{2\;}\;-\;\;2^2\;\;\ast\;\;\left(\sin\left(3^\circ\right)\right)^2}$$

$$a\approx\;1.01$$

Gamma( γ ) = 3
a = 4
b = ?
c = 5

$$b=a\;\ast\;\cos\gamma\;+\;\sqrt{c^2\;-\;a^2\;\;\ast\;\left(\sin\gamma\right)^2}$$

$$b=4\;\ast\;\cos\left(171.89\right)\;+\;\sqrt{5^2\;-\;4^2\;\;\ast\;\left(\sin\left(171.89\right)\right)^2}$$

$$b\approx\;1.01$$

Gamma( γ ) = 3
a = 4
b = 6
c = ?

$$c=\sqrt{a^{2\;}\;+\;\;b^2\;\;-\;\;2\ast a\ast b\;\ast\cos\gamma}$$

$$c=\sqrt{4^{2\;}\;+\;\;6^2\;\;-\;\;2\ast4\ast6\;\ast\cos\left(171.89^\circ\right)}$$

$$c\approx9.97595$$