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Derivatives of the Sine, Cosine and Tangent Functions. by M. Bourne. It can be shown from first principles that: `(d(sin x))/(dx)=cos x` `(d(cos x))/dx=-sin x` `(d(tan x))/(dx)=sec^2x` Explore animations of these functions with their derivatives here: Differentiation Interactive Applet - trigonometric functions.


The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. Common trigonometric functions include sin(x), cos(x) and tan(x). For example, the derivative of f(x) = sin(x) is represented as f ′(a) = cos(a).


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The derivative of sin x. The derivative of cos x. The derivative of tan x. The derivative of cot x. The derivative of sec x. The derivative of csc x. T HE DERIVATIVE of sin x is cos x. To prove that, we will use the following identity: sin A − sin B = 2 cos ½(A + B) sin ½(A − B). (Topic 20 of Trigonometry.) Problem 1. Use that identity to ...


The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin x). The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.


We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. The results are \(\dfrac{d}{dx}\sin x=\cos x\dfrac{d}{dx}\cos x=−\sin x\). With these two formulas, we can determine the derivatives of all six basic trigonometric functions.


sin(x) = d dx sin(x) = cos(x) The derivatives of cos(x) have the same behavior, repeating every cycle of 4. The nth derivative of cosine is the (n+1)th derivative of sine, as cosine is the first derivative of sine. The derivatives of sine and cosine display this cyclic behavior due to their relationship to the complex exponential function.


3. Derivatives of the Inverse Trigonometric Functions. by M. Bourne. Recall from when we first met inverse trigonometric functions: " sin-1 x" means "find the angle whose sine equals x". Example 1. If x = sin-1 0.2588 then by using the calculator, x = 15°. We have found the angle whose sine is 0.2588.


Proofs of the derivative formulas for the sine and cosine functions.