Trigonometric Functions

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 TRIGONOMETRIC FUNCTIONS If we are given we may state that, from the general formula, Since and Then by substituting equations (5.4) and (5.5) into equation (5.3), Now we are interested in finding the derivative of the function sin u, so we apply the chain rule From the chain rule and equation (5.6), we find In other words, to find the derivative of the sine of a function, we use the cosine of the function times the derivative of the function. By a similar process we find the derivative of the cosine func­tion to be The derivatives of the other trigonometric functions may be found by expressing them in terms of the sine and cosine. That is, and by substituting sin u for u, cos u for v, and du for dx in the expression of the quotient theorem we have Taking and and substituting into equation (5.7), we find that Now using the chain rule and equation (5.8), we find By stating the other trigonometric functions in terms of the sine and cosine and using similar processes, we may find the following derivatives: EXAMPLE: Find the derivative of the function SOLUTION. EXAMPLE. Find the derivative of the function SOLUTION: Use the power theorem to find Then find and Combining all of these, we find that PRACTICE PROBLEMS: Find the derivative of the following: ANSWERS: