Theory:

We will derive the trigonometric ratios of \(0^{\circ}\) with the help of a unit circle.
 
A circle with radius \(1\) unit centred at the origin is called a unit circle.
 
0 and 90 deg.png
 
Here \(OQ\) \(=\) \(OP\) \(=\) \(OC\) \(=\) \(1\) unit (Radius).
 
Let us consider the first quadrant.
 
0 and 90 deg illus.png
 
Let the point \(C(x,y)\) be any point on the unit circle and \(\angle COB\) \(=\) \(\theta\).
 
In the right angles triangle \(COB\) we have:
 
Opposite side \(=\) \(y\)
 
Adjacent side \(=\) \(x\)
 
Hypotenuse \(=\) \(1\)
 
Now, let us determine the trigonometric ratios in the first quadrant with the coordinate \(C\).
 
  • Sine \(\theta\):
 
\(\sin \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Hypotenuse}}\)
 
\(=\) \(\frac{y}{1}\)
 
\(=\) \(y\)
 
  • Cosine \(\theta\):
 
\(\cos \theta\) \(=\) \(\frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
 
\(=\) \(\frac{x}{1}\)
 
\(=\) \(x\)
 
  • Tangent \(\theta\):
 
\(\tan \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Adjacent side}}\)
 
\(=\) \(\frac{y}{x}\)
 
When \(\theta = 0^{\circ}\), \(OC\) coincides with \(OP\) then \(P = (1,0)\) where \(x\) \(=\) \(1\) and \(y\) \(=\) \(0\).
 
Then the trigonometric ratios are given by:
 
  • Sine \(0^{\circ}\):
 
\(\sin 0^{\circ}\) \(=\) \(y\)
 
\(=\) \(0\)
 
  • Cosine \(0^{\circ}\):
 
\(\cos 0^{\circ}\) \(=\) \(x\)
 
\(=\) \(1\)
 
  • Tangent \(0^{\circ}\):
 
\(\tan 0^{\circ}\) \(=\) \(\frac{0}{1}\)
 
\(=\) \(0\)
 
Using these basic trigonometric ratios determine their reciprocals as follows:
 
  • Cosecant \(0^{\circ}\):
 
\(\text{cosec}\,0^{\circ}\) \(=\) \(\frac{1}{\sin 0^{\circ}}\)
 
\(=\) \(\frac{1}{0}\)
 
\(=\) not defined
 
  • Secant \(0^{\circ}\):
 
\(\sec 0^{\circ}\) \(=\) \(\frac{1}{\cos 0^{\circ}}\)
 
\(=\) \(\frac{1}{1}\)
 
\(=\) \(1\)
 
  • Cotangent \(0^{\circ}\):
 
\(\cot 0^{\circ}\) \(=\) \(\frac{1}{\tan 0^{\circ}}\)
 
\(=\) \(\frac{1}{0}\)
 
\(=\) not defined
 
Let us summarize all the trigonometric ratios of \(0^{\circ}\) in the following table.
 
 
\(\sin \theta\)
\(\cos \theta\)
\(\tan \theta\)
\(\text{cosec}\,\theta\)
\(\sec \theta\)
\(\cot \theta\)
\(\theta = 0^{\circ}\)
\(0\)
\(1\)
\(0\)
not defined
 \(1\)
not defined