Law of Cosines

The law of cosines for calculating one side of a triangle when the angle opposite and the other two sides are known. Can be used in conjunction with the law of sines to find all sides and angles.
Side a=
Side b=
Side c=
Angle C= degrees
Enter data for sides a and b and either side c or angle C. Then click on the active text for the unknown quantity you wish to calculate.
Calculation of side c is straightforward, but calculation of sides a or b is more involved since changing either side c or angle C forces changes in both a and b. To calculate a or b, first use the law of sines to find the angle opposite the side you wish to calculate. Then use the law of cosines to find the unknown side length.

Applications

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Applications, Law of Cosines

The law of cosines has application to vector quantities:
To find the difference between two vectors, asin a glancing collision.

It has application along with the law of sines to the problem of the heading angle for an aircraft in the wind.

Other applications:

Lande' g-factor
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Aircraft Heading to Counter Wind

Calculating the necessary aircraft heading to counter a wind velocity and proceed along a desired bearing to a destination is a classic problem in aircraft navigation. It makes good use of the law of sines and the law of cosines.

The angle θ is justthe difference between the wind direction and thedesired bearing. With that angle and the law of sines,the offset angle β for theaircraft is obtained:


Then the law of cosines is applied to get the resultant ground speed:

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