Laws of Sines and Cosines

The Laws of Sines and Cosines are useful for geometry questions involving non-right triangles.

Law of Sines

The Law of Sines states that 

aSin(A)=bSin(B)=cSin(C) where a, b, and c are the side lengths of the triangle, and A, B, and C are the internal angles opposite those respective sides, respectively.

Questions involving the Law of Sines or Law of Cosines will not normally involve calculating a specific number. Instead, it will often be about plugging numbers into the equation, and then isolating a variable in that equation.

Here is a sample question using the equation:

This is a perimeter question. Since there needs to be a firefighter every 4 meters, the perimeter will need to be divided by 4 to calculate the number of firefighters needed. This eliminates answer C, since it is the only answer not divided by 4.

Answers D and E can also be eliminated, since they utilize an area equation rather than calculating a perimeter.

So the answer must be either A or B. This also makes sense, since the perimeter sum should include 130 meters, as well as the lengths of the other two sides (which will be calculated using the law of sines, so having the two angle ratios for the other two sides makes sense).

For this triangle, the Law of Sines would look like: 

130sin(91) =bsin(47)=csin(42) .

So the lengths of b or c are equal to 

130sin(91)multiplied by the sin of their opposite angles. So the correct equation should have sin(91) in both denominators and then sin(42) and sin(47) multiplied by 130 in each numerator. So the correct answer is A.

Law of Cosines

The Law of Cosines is similar to the Law of Sines, but rather than comparing individual ratios of each side/angle pair it includes all:

c²= a² + b² − (2ab • cos(C)).