Learning Objectives
- 3.5.1 Find the derivatives of the sine and cosine function.
- 3.5.2 Find the derivatives of the standard trigonometric functions.
- 3.5.3 Calculate the higher-order derivatives of the sine and cosine.
One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.
Derivatives of the Sine and Cosine Functions
We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function
Consequently, for values of very close to 0, We see that by using
By setting and using a graphing utility, we can get a graph of an approximation to the derivative of (Figure 3.25).
Upon inspection, the graph of appears to be very close to the graph of the cosine function. Indeed, we will show that
If we were to follow the same steps to approximate the derivative of the cosine function, we would find that
The Derivatives of sin x and cos x
The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.
Proof
Because the proofs for and use similar techniques, we provide only the proof for Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:
The graphs of and are shown in Figure 3.26.
We also recall the following trigonometric identity for the sine of the sum of two angles:
Now that we have gathered all the necessary equations and identities, we proceed with the proof.
□
Figure 3.27 shows the relationship between the graph of and its derivative Notice that at the points where has a horizontal tangent, its derivative takes on the value zero. We also see that where is increasing, and where is decreasing,
Example 3.39
Differentiating a Function Containing sin x
Find the derivative of
Checkpoint 3.25
Find the derivative of
Example 3.40
Finding the Derivative of a Function Containing cos x
Find the derivative of
Checkpoint 3.26
Find the derivative of
Example 3.41
An Application to Velocity
A particle moves along a coordinate axis in such a way that its position at time is given by for At what times is the particle at rest?
Checkpoint 3.27
A particle moves along a coordinate axis. Its position at time is given by for At what times is the particle at rest?
Derivatives of Other Trigonometric Functions
Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.
Example 3.42
The Derivative of the Tangent Function
Find the derivative of
Find the derivative of
The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.
Derivatives of and
The derivatives of the remaining trigonometric functions are as follows:
Example 3.43
Finding the Equation of a Tangent Line
Find the equation of a line tangent to the graph of at
Example 3.44
Finding the Derivative of Trigonometric Functions
Find the derivative of
Find the derivative of
Find the slope of the line tangent to the graph of at
Higher-Order Derivatives
The higher-order derivatives of and follow a repeating pattern. By following the pattern, we can find any higher-order derivative of and
Example 3.45
Finding Higher-Order Derivatives of
Find the first four derivatives of
Analysis
Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of sin x equals sin x, so
For find
Example 3.46
Using the Pattern for Higher-Order Derivatives of
Find
For find
Example 3.47
An Application to Acceleration
A particle moves along a coordinate axis in such a way that its position at time is given by Find and Compare these values and decide whether the particle is speeding up or slowing down.
A block attached to a spring is moving vertically. Its position at time is given by Find and Compare these values and decide whether the block is speeding up or slowing down.
Section 3.5 Exercises
For the following exercises, find for the given functions.
For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.
[T]
[T]
[T]
For the following exercises, find for the given functions.
Find all values on the graph of for where the tangent line has slope 2.
Let Determine the points on the graph of for where the tangent line(s) is (are) parallel to the line
[T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function where is measured in inches and is measured in seconds. Find the rate at which the spring is oscillating at s.
Let the position of a swinging pendulum in simple harmonic motion be given by where and are constants, measures time in seconds, and measures position in centimeters. If the position is 0 cm and the velocity is 3 cm/s when , find the values of and .
After a diver jumps off a diving board, the edge of the board oscillates with position given by cm at seconds after the jump.
- Sketch one period of the position function for
- Find the velocity function.
- Sketch one period of the velocity function for
- Determine the times when the velocity is 0 over one period.
- Find the acceleration function.
- Sketch one period of the acceleration function for
The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by where is the number of hamburgers sold and represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find and determine the intervals where the number of burgers being sold is increasing.
[T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by where is months since January. Find and use a calculator to determine the intervals where the amount of rain falling is decreasing.
For the following exercises, use the quotient rule to derive the given equations.
Use the definition of derivative and the identity
to prove that
For the following exercises, find the requested higher-order derivative for the given functions.
of
of