April 2009

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 $dA/dx = 0$.
 

Graph of the Function y = f(x)
The graph of a function y = f(x) may be plotted using Differential Calculus. Consider the graph shown below.
 

As x increases, the curve rises if the slope is positive, as of arc AB; it falls if the slope is negative, as of arc BC.
 

The sum and difference of two angles can be derived from the figure shown below.