In calculus the relations stated below require angles to be measured in radians; the relations would become more complicated if angles were measured in another unit such as degrees. If the trigonometric functions are defined in terms of geometry, their derivatives can be found by verifying two limits. The first is:

verified using the unit circle and squeeze theorem. It may be tempting to propose to use L'Hôpital's rule to establish this limit. However, if one uses this limit in order to prove that the derivative of the sine is the cosine, and then uses the fact that the derivative of the sine is the cosine in applying L'Hôpital's rule, one is reasoning circularly—a logical fallacy. The second limit is:

verified using the identity tan("x"/2) = (1 − cos("x"))/sin("x"). Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that sin′("x") = cos("x") and cos′("x") = −sin("x"). If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.

The rest of the trigonometric functions can be differentiated using the above identities and the rules of derivative:

The integral identities can be found in "list of integrals of trigonometric functions".

आवर्तता (periodicity)
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चलन-कलन (Calculus)
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