Law of Sines and Cosines

Sub Topics
The law of sines and cosines can be used to find the sides of the traingle with one side and two angles given or two sides and one angle given. This laws is in convention with the normal ones that we use in geometry.

Law of sines and cosines Formula

The sine ratio and cosine ratio of any angle is an information which is readily available. In addition, we know the identity that sum of all interior angles in any triangle is always 180o. These two known facts used in interesting applications in solving any triangle.

Firstly, with the known facts we are in a position to derive two important formulas that connect the sine values/cosine values of angles of any triangle with the measures of sides of the same triangle. Thus basically, laws of sine and cosine are the formulas that is related to the study of any type of triangles.

Now let us consider a triangle general in nature, that is a scalene triangle which has all interior angle different and all measures of the sides are different as well. Such a triangle is shown below.

In any scalene triangle ABC , as shown above,the law of sine in the form of formula is,
$[\frac{(a)}{\sin A}]$ = $[\frac{(b)}{(\sin B)}]$ = $[\frac{(c)}{\sin C}]$And the law of cosine in the form of formula is,

a2 = b2 + c2 - 2bc cos A,   b2 = c2 + a2 - 2ca cos B   and    c2 = a2 + b2 - 2ab cos CIt may be noted that both law of sine and law of cosine relates the angles and the sides. In other words, the with the minimal information of these parameters, all the remaining parameters can be found.

It may appear that it is easier to work with law of sine but the law of sine can not be used when the measures of all three sides are known but none of the angles are known. In such a case, the law of cosine is the only source of help, till at least finding one of the angles.

Similarly law of cosine is ineffective if we only know the measures of two sides and one angle which is not an included angle. In this case, the entire triangle can be solved by law of sine alone. But in such a case, there is a possibility of getting two solutions or ambiguous solutions.  Both the  solutions may be correct or one of them may be extraneous depending upon the application. Let us explain cases where two possible solutions can occur.

Consider the same diagram for a case when the measures of sides b, c and measure of angle B is known. There will be two solutions for angle A correspondingly two solutions for the measures of side a, if,

1) angle B is acute
2) b < c
3) b > c $\times$ sin B

Similarly the law of cosine is also helpful to find the type of triangle when you know only the measures of all the sides. The method is, label the greatest side as a, thereby, the greatest angle is A. Now rewriting the formula  a2 = b2 + c2 - 2bc cos A as,
$\cos A$ = $\frac{[(b^{2}+c^{2})-(a^{2})]}{[(2bc)]}$If cos A is positive, the angle A is acute, and hence the triangle is an acute triangle as A is the greatest angle. If cos A is negative, the angle A is obtuse, and hence the triangle is an obtuse triangle as A is the greatest angle. If cos A is 0, the angle A is right angle, and hence the triangle is a right triangle.

Law of Sines and Cosines Word Problems

Below you could see law of sines and cosines word problems