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2.2. THE SINE AND COSINE FUNCTIONS
2.2 The sine and cosine functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a graphical justification for why
d
dx [a x ] = a x ln(a)?
• What do the graphs of y = sin(x) and y = cos(x) suggest as formulas for their
respective derivatives?
• Once we know the derivatives of sin(x) and cos(x), how do previous derivative
rules work when these functions are involved?
Introduction
Throughout Chapter 2, we will be working to develop shortcut derivative rules that will
help us to bypass the limit definition of the derivative in order to quickly determine the
formula for f ′ (x) when we are given a formula for f (x). In Section 2.1, we learned the
rule for power functions, that if f (x) = x n , then f ′ (x) = nx n−1 , and justified this in part
due to results from different n-values when applying the limit definition of the derivative.
We also stated the rule for exponential functions, that if a is a positive real number and
f (x) = a x , then f ′ (x) = a x ln(a). Later in this present section, we are going to work
to conjecture formulas for the sine and cosine functions, primarily through a graphical
argument. To help set the stage for doing so, the following preview activity asks you to
think about exponential functions and why it is reasonable to think that the derivative of
an exponential function is a constant times the exponential function itself.
Preview Activity 2.2. Consider the function g(x) = 2 x , which is graphed in Figure 2.2.
(a) At each of x = −2, −1, 0, 1, 2, use a straightedge to sketch an accurate tangent line
to y = g(x).
(b) Use the provided grid to estimate the slope of the tangent line you drew at each
point in (a).
(c) Use the limit definition of the derivative to estimate g ′ (0) by using small values of
h, and compare the result to your visual estimate for the slope of the tangent line
to y = g(x) at x = 0 in (b).
(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) Write at least one sentence that explains why it is reasonable to think that
g ′ (x) = cg(x), where c is a constant. In addition, calculate ln(2), and then
2.2. THE SINE AND COSINE FUNCTIONS
2.2 The sine and cosine functions
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a graphical justification for why
d
dx [a x ] = a x ln(a)?
• What do the graphs of y = sin(x) and y = cos(x) suggest as formulas for their
respective derivatives?
• Once we know the derivatives of sin(x) and cos(x), how do previous derivative
rules work when these functions are involved?
Introduction
Throughout Chapter 2, we will be working to develop shortcut derivative rules that will
help us to bypass the limit definition of the derivative in order to quickly determine the
formula for f ′ (x) when we are given a formula for f (x). In Section 2.1, we learned the
rule for power functions, that if f (x) = x n , then f ′ (x) = nx n−1 , and justified this in part
due to results from different n-values when applying the limit definition of the derivative.
We also stated the rule for exponential functions, that if a is a positive real number and
f (x) = a x , then f ′ (x) = a x ln(a). Later in this present section, we are going to work
to conjecture formulas for the sine and cosine functions, primarily through a graphical
argument. To help set the stage for doing so, the following preview activity asks you to
think about exponential functions and why it is reasonable to think that the derivative of
an exponential function is a constant times the exponential function itself.
Preview Activity 2.2. Consider the function g(x) = 2 x , which is graphed in Figure 2.2.
(a) At each of x = −2, −1, 0, 1, 2, use a straightedge to sketch an accurate tangent line
to y = g(x).
(b) Use the provided grid to estimate the slope of the tangent line you drew at each
point in (a).
(c) Use the limit definition of the derivative to estimate g ′ (0) by using small values of
h, and compare the result to your visual estimate for the slope of the tangent line
to y = g(x) at x = 0 in (b).
(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) Write at least one sentence that explains why it is reasonable to think that
g ′ (x) = cg(x), where c is a constant. In addition, calculate ln(2), and then