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Derivatives of Trigonometric Functions

We now give one way of calculating the derivative of the sine function. Let f (x) = sin(x). Using the trigonometric identity sin(a + b) = sin(a)cos(b) + sin(b)cos(a), we have


f'(x)=  
 =  
 =  
 =sin(x) + cos(x)  
 =cos(x)  

where the last equality follows from examining the figure below:

Figure %: Calculating the Derivative of the Sine Function

We may similarly compute the derivative of g(x) = cos(x) to be g'(x) = - sin(x). Finally, since tan(x) = sin(x)/cos(x), it will follow from the quotient rule that the derivative of h(x) = tan(x) is h'(x) = 1/(cos(x))2.

We will compute the derivatives of the inverse trigonometric functions in the next section, using implicit differentiation.

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