Chapter 2: Differentiation
Section 2.7: Derivatives of the Trig Functions
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Essentials
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Introduction
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Table 2.7.1 lists the derivatives of the six trigonometric functions.
Function
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Maple's Derivative
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Textbook Derivative
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sine
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=
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cosine
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=
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tangent
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=
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cotangent
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=
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secant
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=
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cosecant
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=
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Table 2.7.1 Derivatives of the six trigonometric functions
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Maple returns the derivatives of the tangent and cotangent functions in a form that differs from the form found in the typical calculus textbook. (Note the red cells in Table 2.7.1.) This is because early-on, the Maple programmers opined that for these two functions, returning the derivative in terms of the same function was somehow "simpler" than returning it in terms of a different function. Table 2.7.2 contains Maple code for modifying the differentiation rules for these two functions. (The restart is in deference to any remember-table issues.)
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restart;
unprotect(`diff/tan`):
unprotect(`diff(cot`):
`diff/tan` := proc(a,x) sec(a)^2*diff(a,x);end:
`diff/cot` := proc(a,x) -csc(a)^2*diff(a,x);end:
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Table 2.7.2 Code to modify the differentiation rules for tangent and cotangent
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If the differentiation rules are arranged as in Table 2.7.3, certain relations between all the trig functions and their derivatives can be observed.
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csc
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Table 2.7.3 Derivatives of the trigonometric functions
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If each function in the left-hand column is replaced by its co-function, and a minus sign inserted for the derivative, the right-hand column results.
The derivatives of the sine and cosine functions follow from an application of Definition 2.2.1, the definition of the derivative, and from the two special limits in Table 1.4.1. The derivatives of the remaining functions are obtained by applying the Quotient, Power, and Chain rules to the forms of the functions shown in Table 2.7.4.
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Table 2.7.4 Expressing trig functions in terms of sine and cosine
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Derivative of the Sine Function
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A graph of the sine function suggests that this function is continuous and has a tangent line at all points. This suggests that the sine function should be differentiable everywhere.
Figure 2.7.1 shows the result of applying the
to the sine function.
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Figure 2.7.1 Application of the Derivative Tutor to the function
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The red curve is the graph of , whereas the blue curve is the graph of its derivative. This blue curve seems to be periodic with period , and amplitude 1. Moreover, it seems to be a translation of the red curve, that is, a translation by of . This suggests that the derivative of the sine function might indeed be the cosine function. In Table 2.7.5, an application of Definition 2.2.1 verifies that this conjecture is correct.
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Table 2.7.5 Application of Definition 2.2.1 to the sine function
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Table 2.7.6 lists the higher derivatives of the sine function.
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<<k,seq(k,k=0..8)>|<d/dx^k,seq(diff(sin(x),[x$k]),k=0..8)>>;
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Table 2.7.6 Higher derivatives of
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The convention is that = , that is, the zeroth-ordered derivative is the function itself since no derivative is taken. The table implies that the higher-order derivatives of the sine function are cyclic, with period 4.
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Derivative of the Cosine Function
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A graph of the cosine function suggests that this function is continuous and has a tangent line at all points. This suggests that the cosine function should be differentiable everywhere.
Figure 2.7.2 shows the result of applying the
to the cosine function.
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Figure 2.7.2 Application of the Derivative Tutor to the function
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The red curve is the graph of , whereas the blue curve is the graph of its derivative. This blue curve seems to be periodic with period , and amplitude 1. Moreover, it seems to be a translation of the red curve, that is, a translation by of . This suggests that the derivative of the cosine function might indeed be the sine function. In Table 2.7.7, an application of Definition 2.2.1 verifies that this conjecture is correct.
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Table 2.7.7 Application of Definition 2.2.1 to the cosine function
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Table 2.7.8 lists the higher derivatives of the sine function.
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<<k,seq(k,k=0..8)>|<d/dx^k,seq(diff(cos(x),[x$k]),k=0..8)>>;
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Table 2.7.8 Higher derivatives of
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The convention is that = , that is, the zeroth-ordered derivative is the function itself since no derivative is taken. The table implies that the higher-order derivatives of the cosine function are cyclic, with period 4.
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Derivative of the Tangent Function
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Table 2.7.9 contains the calculations by which the derivative of the tangent function is obtained. The differentiation begins with an application of the Quotient rule.
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Table 2.7.9 Derivative of the tangent function
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Derivative of the Cotangent Function
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Table 2.7.10 contains the calculations by which the derivative of the cotangent function is obtained. The differentiation again begins with an application of the Quotient rule.
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Table 2.7.10 The derivative of the cotangent function
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Derivative of the Secant Function
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Table 2.7.11 contains the calculations by which the derivative of the secant function is obtained. The differentiation begins with an application of the Power rule, followed by an application of the Chain rule.
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Table 2.7.11 Derivative of the secant function
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Derivative of the Cosecant Function
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Table 2.7.12 contains the calculations by which the derivative of the cosecant function is obtained. The differentiation again begins with an application of the Power rule, followed by an application of the Chain rule.
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Table 2.7.12 Derivative of the cosecant function
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Examples
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