UPSKILL MATH PLUS

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We will derive the trigonometric ratios of \(0^{\circ}\) using the right angled triangle \(ABC\).
 
0_Deg.png
 
Let us now experiment with the given triangle concerning \(\angle B\).
 
Reduce \(\theta\) to the extent it becomes zero.
 
0_Deg.gif
 
It is observed that as \(\angle B\) gets smaller and smaller, the point \(C\) gets closer to the point \(A\).
 
That is, when \(\theta\) becomes very close to \(0^{\circ}\), the side \(BC\) becomes at most the same as the side \(AB\).
 
This implies that the measure of \(AC\) becomes almost zero.
 
In the right angles triangle \(ABC\) we have:
 
Opposite side \(=\) \(AC\)
 
Adjacent side \(=\) \(AB\)
 
Hypotenuse \(=\) \(BC\)
 
Now, let us determine the trigonometric ratios when \(\theta = 0^{\circ}\) as follows.
 
  • Sine \(0^{\circ}\):
 
\(\sin \theta\) \(=\) \(\frac{\text{Opposite side}}{\text{Hypotenuse}}\)
 
\(\sin \theta\) \(=\) \(\frac{AC}{BC}\)
 
\(\sin 0^{\circ}\) \(=\) \(\frac{0}{BC}\)
 
\(=\) \(0\)
 
  • Cosine \(0^{\circ}\):
 
\(\cos \theta\) \(=\) \(\frac{\text{Adjacent side}}{\text{Hypotenuse}}\)
 
\(\cos \theta\) \(=\) \(\frac{AB}{BC}\)
 
\(\cos 0^{\circ}\)\(=\) \(1\) [When \(\angle B = 0^{\circ}\), \(AB\) \(=\) \(BC\)]
 
  • Tangent \(0^{\circ}\):
 
\(\tan 0^{\circ}\) \(=\) \(\frac{\sin 0^{\circ}}{\cos 0^{\circ}}\)
 
\(=\) \(\frac{0}{1}\)
 
\(=\) \(0\)
 
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