To evaluate the six trigonometric functions at \(0\) radians (or \(0\) degrees), we can find the following:
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Sine: \[ \sin(0) = 0 \]
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Cosine: \[ \cos(0) = 1 \]
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Tangent: \[ \tan(0) = \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 \]
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Cosecant (the reciprocal of sine): \[ \csc(0) = \frac{1}{\sin(0)} = \frac{1}{0} \] Note: \(\csc(0)\) is undefined because you cannot divide by zero.
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Secant (the reciprocal of cosine): \[ \sec(0) = \frac{1}{\cos(0)} = \frac{1}{1} = 1 \]
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Cotangent (the reciprocal of tangent): \[ \cot(0) = \frac{1}{\tan(0)} = \frac{1}{0} \] Note: \(\cot(0)\) is also undefined because you cannot divide by zero.
In summary, the values of the six trigonometric functions at \(0\) are:
- \(\sin(0) = 0\)
- \(\cos(0) = 1\)
- \(\tan(0) = 0\)
- \(\csc(0) = \text{undefined}\)
- \(\sec(0) = 1\)
- \(\cot(0) = \text{undefined}\)