Sine and Cosine Rules

As with all other mathematics, there has to be applications of trigonometry. Here are the 3 basic problems central to navigation, architecture, and physics:

1) If we knew the length of 1 side and any 2 angles of the triangle, could we find the last angle and the length of the other 2 sides?
2) If we knew the length of any 2 sides and the angle in between the two sides of the triangle, could we find the other two angles and the length of the other side?
3) If we knew the length of all 3 sides of the triangle, could we find all 3 angles?

Try to come up with a method to these problems. Is it possible to work out the values without actually drawing an accurate diagram?

Mathematicians have again worked out 2 important observations regarding the above problems. They are known as the sine rule and cosine rule.

Note: In the proofs below, we chose only 1 vertex to drop the perpendicular from. In fact, any of the 3 would do, and we generally use all 3 to get the complete observation. However, we would only go through the proof once, and the reasoning is the same for the other 2 vertices.

Sine Rule:



In the diagram above, we drop a perpendicular from C onto AB, and label that point D. We will find h in terms of A, a, B and b in 2 different ways:

As with the trigonometry of right angles,

Triangle BCD: sin(B)=h/a
Triangle ACD: sin(A)=h/b

Then, h=asin(B)=bsin(A). In conclusion, we have a/sin(A)=b/sin(B)=c/sin(C).

Cosine Rule:



In the diagram above, we drop the perpendicular from B onto AC, and label that point D. The lengths are defined as in the diagram.

By applying Pythagoras's Theorem,

Triangle ABD: h^2=c^2-n^2
Triangle BCD: h^2=a^2-(b-n)^2

Combining them together and eliminating h^2, we get c^2-a^2=n^2-(b-n)^2=n^2-(b^2-2bn+n^2)=2bn-b^2.

Then, applying right-angle trigonometry on triangle ABD, we get cos(A)=n/c, or n=ccos(A). Substituting this to the above gives us c^2-a^2=2bccosA-b^2, or a^2=b^2+c^2-2bccosA.

Now, try the problems that follow:

1) We label the lengths of the sides of the triangle opposite vertices A, B and C be a, b and c respectively, and the angles A, B and C.
a) Find the values of b, c and C if a=6, A=30 degrees and B=70 degrees.
b) Find the values of A, B and C if a=4, b=7 and c=8.

2) What is the minimum number of the variables A, B, C, a, b and c do we need to know the values of do we need to find the rest of them? Prove your conjecture.

3) Find the possible values of A, C and c given that a=5, b=7 and B=60 degrees. Why are there 2 solutions?

4) Let AD be an angle bisector of angle BAC in triangle ABC. D lies on BC. Show that AB*CD=AC*BD. This is known as the Angle Bisector Theorem.

5) In triangle ABC, the sides of the triangle have lengths x-1, x and x+1. If the largest angle is twice of the smaller angle, find the perimeter of the triangle.