2.2. THE SINE AND COSINE FUNCTIONS
99
(d) Based on your work in (a), (b), and (c), sketch an accurate graph of y = g ′ (x)
on the axes adjacent to the graph of y = g(x).
(e) What familiar function do you think is the derivative of g(x) = cos(x)?
⊳
The results of the two preceding activities suggest that the sine and cosine functions
not only have the beautiful interrelationships that are learned in a course in trigonometry
– connections such as the identities sin
2 (x) + cos 2 (x) = 1 and cos(x −
π
2 ) = sin(x) – but
that they are even further linked through calculus, as the derivative of each involves the
other. The following rules summarize the results of the activities 4 .
Sine and Cosine Functions: For all real numbers x,
d
dx
[sin(x)] = cos(x) and
d
dx
[cos(x)] = − sin(x)
We have now added two additional functions to our library of basic functions whose
derivatives we know: power functions, exponential functions, and the sine and cosine
functions. The constant multiple and sum rules still hold, of course, and all of the inherent
meaning of the derivative persists, regardless of the functions that are used to constitute a
given choice of f (x). The following activity puts our new knowledge of the derivatives of
sin(x) and cos(x) to work.
Activity 2.6.
Answer each of the following questions. Where a derivative is requested, be sure to
label the derivative function with its name using proper notation.
(a) Determine the derivative of h(t) = 3 cos(t) − 4 sin(t).
(b) Find the exact slope of the tangent line to y = f (x) = 2x +
sin(x)
2
at the point
where x =
π
6 .
(c) Find the equation of the tangent line to y = g(x) = x 2 + 2 cos(x) at the point
where x =
π
2 .
(d) Determine the derivative of p(z) = z 4 + 4 z + 4 cos(z) − sin(
π
2 ).
(e) The function P(t) = 24 + 8 sin(t) represents a population of a particular kind
of animal that lives on a small island, where P is measured in hundreds and t
is measured in decades since January 1, 2010. What is the instantaneous rate of
change of P on January 1, 2030? What are the units of this quantity? Write a
sentence in everyday language that explains how the population is behaving at
this point in time.
4 These two rules may be formally proved using the limit definition of the derivative and the expansion
identities for sin(x + h) and cos(x + h).
99
(d) Based on your work in (a), (b), and (c), sketch an accurate graph of y = g ′ (x)
on the axes adjacent to the graph of y = g(x).
(e) What familiar function do you think is the derivative of g(x) = cos(x)?
⊳
The results of the two preceding activities suggest that the sine and cosine functions
not only have the beautiful interrelationships that are learned in a course in trigonometry
– connections such as the identities sin
2 (x) + cos 2 (x) = 1 and cos(x −
π
2 ) = sin(x) – but
that they are even further linked through calculus, as the derivative of each involves the
other. The following rules summarize the results of the activities 4 .
Sine and Cosine Functions: For all real numbers x,
d
dx
[sin(x)] = cos(x) and
d
dx
[cos(x)] = − sin(x)
We have now added two additional functions to our library of basic functions whose
derivatives we know: power functions, exponential functions, and the sine and cosine
functions. The constant multiple and sum rules still hold, of course, and all of the inherent
meaning of the derivative persists, regardless of the functions that are used to constitute a
given choice of f (x). The following activity puts our new knowledge of the derivatives of
sin(x) and cos(x) to work.
Activity 2.6.
Answer each of the following questions. Where a derivative is requested, be sure to
label the derivative function with its name using proper notation.
(a) Determine the derivative of h(t) = 3 cos(t) − 4 sin(t).
(b) Find the exact slope of the tangent line to y = f (x) = 2x +
sin(x)
2
at the point
where x =
π
6 .
(c) Find the equation of the tangent line to y = g(x) = x 2 + 2 cos(x) at the point
where x =
π
2 .
(d) Determine the derivative of p(z) = z 4 + 4 z + 4 cos(z) − sin(
π
2 ).
(e) The function P(t) = 24 + 8 sin(t) represents a population of a particular kind
of animal that lives on a small island, where P is measured in hundreds and t
is measured in decades since January 1, 2010. What is the instantaneous rate of
change of P on January 1, 2030? What are the units of this quantity? Write a
sentence in everyday language that explains how the population is behaving at
this point in time.
4 These two rules may be formally proved using the limit definition of the derivative and the expansion
identities for sin(x + h) and cos(x + h).