The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.
 

For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...
 

Half angle formulas can be derived from the double angle formulas, particularly, the cosine of double angle. For easy reference, the cosines of double angle are listed below:
 

Note that the equations above are identities, meaning, the equations are true for any value of the variable θ. The key on the derivation is to substitute θ with ½ θ.
 

The derivation of basic identities can be done easily by using the functions of a right triangle. For easy reference, these trigonometric functions are listed below.

a/c = sin θ
b/c = cos θ
a/b = tan θ
c/a = csc θ
c/b = sec θ
b/a = cot θ

From the right triangle shown below,
 

the trigonometric functions of angle θ are defined as follows:

Basic Identities
See the derivation of basic identities.

1. $\sin \theta = \dfrac{1}{\csc \theta} ~ \Leftrightarrow ~ \csc \theta = \dfrac{1}{\sin \theta}$
    
2. $\cos \theta = \dfrac{1}{\sec \theta} ~ \Leftrightarrow ~ \sec \theta = \dfrac{1}{\cos \theta}$
    
3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta} ~ \Leftrightarrow ~ \cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$

In reference to the right triangle shown and from the functions of a right triangle:
a/c = sin θ
b/c = cos θ
c/b = sec θ
c/a = csc θ
a/b = tan θ
b/a = cot θ
 

As an example, the area of a rectangular lot, expressed in terms of its length and width, may also be expressed in terms of the cost of fencing. Thus the area can be expressed as $A = f(x)$. The common task here is to find the value of $x$ that will give a maximum value of $A$. To find this value, we set $dA/dx = 0$.