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We will derive the trigonometric ratios of \(0^{\circ}\) using the right angled triangle \(ABC\).
Let us now experiment with the given triangle concerning \(\angle B\).
Reduce \(\theta\) to the extent it becomes zero.
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 |
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