trigonometry basics formula

Area of Triangle (Proof)

 [1.1]
In the triangle below, the height is h. The area is:
  [1.2]
(Half the base times the height, of course)

h=b·sinC  [1.3]
So substituting in 1.2, we have:

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Sine Rule (Proof)

Re-using the above triangle, in triangle AXC, 
h/b=sin A
h=b·sin A  [2.1]

In triangle XBC, 
h/a=sin B
h=a·sin B [2.2]

Equating Equations 2.1 and 2.2, we have
h=b·sin A=a·sin B
So, 
b/sin B=a/sin A

Using a perpendicular from A to BC, we can show that 
b/sin B=c/sin C
Hence we have the Sine Rule:
 [2.3]
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Cosine Rule (Proof)

The Cosine Rule is:

 

To prove it we use the triangle below:


h is the height (CX) and x is the distance AX, and, because AB=c, then XB=c-x

In triangle AXC, by Pythagoras' Theorem:
b²=h²+x² 
h²=b²−x²   [3.2]

In triangle XBC, by Pythagoras' Theorem:

a²=(c-x)²+h² 
a²=c²+x²-2cx+h²  

Substituting the value for h² in Equation 3.2 in Equation 3.4:


a²=c²+x²−2cx+b²−x²   

The x2 cancels and by slight rearranging:
a²=b²+ c²−2cx 

In triangle AXC, we note:
x=b·cosA   

Using this value in Equation 3.6, we get the Cosine Rule: