Study Guide - Derivatives of Trigonometric Functions (2024)

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 [latex]f(x),[/latex]

[latex]f^{\prime}(x)=\underset{h\to 0}{\lim}\frac{f(x+h)-f(x)}{h}[/latex].

Consequently, for values of [latex]h[/latex] very close to 0, [latex]f^{\prime}(x)\approx \frac{f(x+h)-f(x)}{h}[/latex]. We see that by using [latex]h=0.01[/latex],

[latex]\frac{d}{dx}(\sin x)\approx \frac{\sin(x+0.01)-\sin x}{0.01}[/latex]

By setting [latex]D(x)=\frac{\sin(x+0.01)-\sin x}{0.01}[/latex] and using a graphing utility, we can get a graph of an approximation to the derivative of [latex] \sin x[/latex] ((Figure)).

Study Guide - Derivatives of Trigonometric Functions (1) Figure 1. The graph of the function [latex]D(x)[/latex] looks a lot like a cosine curve.

Upon inspection, the graph of [latex]D(x)[/latex] appears to be very close to the graph of the cosine function. Indeed, we will show that

[latex]\frac{d}{dx}(\sin x)= \cos x[/latex].

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that

[latex]\frac{d}{dx}(\cos x)=−\sin x[/latex].

The Derivatives of [latex]\sin x[/latex] and [latex]\cos x[/latex]

The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.

[latex]\frac{d}{dx}(\sin x)= \cos x[/latex]

[latex]\frac{d}{dx}(\cos x)=−\sin x[/latex]

Proof

Because the proofs for [latex]\frac{d}{dx}(\sin x)= \cos x[/latex] and [latex]\frac{d}{dx}(\cos x)=−\sin x[/latex] use similar techniques, we provide only the proof for [latex]\frac{d}{dx}(\sin x)= \cos x[/latex]. Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:

[latex]\underset{h\to 0}{\lim}\frac{\sin h}{h}=1[/latex] and [latex]\underset{h\to 0}{\lim}\frac{\cos h-1}{h}=0[/latex].

The graphs of [latex]y=\frac{\sin h}{h}[/latex] and [latex]y=\frac{(\cos h-1)}{h}[/latex] are shown in (Figure).

Study Guide - Derivatives of Trigonometric Functions (2) Figure 2. These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.

We also recall the following trigonometric identity for the sine of the sum of two angles:

[latex] \sin(x+h)= \sin x \cos h+ \cos x \sin h[/latex].

Now that we have gathered all the necessary equations and identities, we proceed with the proof.

[latex]\begin{array}{lllll}\frac{d}{dx} \sin x & =\underset{h\to 0}{\lim}\frac{\sin(x+h)-\sin x}{h} & & & \text{Apply the definition of the derivative.} \\ & =\underset{h\to 0}{\lim}\frac{\sin x \cos h+ \cos x \sin h- \sin x}{h} & & & \text{Use trig identity for the sine of the sum of two angles.} \\ & =\underset{h\to 0}{\lim}(\frac{\sin x \cos h-\sin x}{h}+\frac{\cos x \sin h}{h}) & & & \text{Regroup.} \\ & =\underset{h\to 0}{\lim}(\sin x(\frac{\cos h-1}{h})+ \cos x(\frac{\sin h}{h})) & & & \text{Factor out} \, \sin x \, \text{and} \, \cos x. \\ & = \sin x(0)+ \cos x(1) & & & \text{Apply trig limit formulas.} \\ & = \cos x & & & \text{Simplify.} \end{array} _\blacksquare[/latex]

(Figure) shows the relationship between the graph of [latex]f(x)= \sin x[/latex] and its derivative [latex]f^{\prime}(x)= \cos x[/latex]. Notice that at the points where [latex]f(x)= \sin x[/latex] has a horizontal tangent, its derivative [latex]f^{\prime}(x)= \cos x[/latex] takes on the value zero. We also see that where [latex]f(x)= \sin x[/latex] is increasing, [latex]f^{\prime}(x)= \cos x>0[/latex] and where [latex]f(x)= \sin x[/latex] is decreasing, [latex]f^{\prime}(x)= \cos x<0[/latex].

Study Guide - Derivatives of Trigonometric Functions (3) Figure 3. Where [latex]f(x)[/latex] has a maximum or a minimum, [latex]f^{\prime}(x)=0[/latex]. That is, [latex]f^{\prime}(x)=0[/latex] where [latex]f(x)[/latex] has a horizontal tangent. These points are noted with dots on the graphs.

Differentiating a Function Containing [latex]\sin x[/latex]

Find the derivative of [latex]f(x)=5x^3 \sin x[/latex].

Answer: Using the product rule, we have

[latex]\begin{array}{ll}f^{\prime}(x) & =\frac{d}{dx}(5x^3)\cdot \sin x+\frac{d}{dx}(\sin x)\cdot 5x^3 \\ & =15x^2\cdot \sin x+ \cos x\cdot 5x^3\end{array}[/latex]

After simplifying, we obtain

[latex]f^{\prime}(x)=15x^2 \sin x+5x^3 \cos x[/latex].

Find the derivative of [latex]f(x)= \sin x \cos x.[/latex]

Answer:

[latex]f^{\prime}(x)=\cos^2 x-\sin^2 x[/latex]

Hint

Don’t forget to use the product rule.

Finding the Derivative of a Function Containing [latex]\cos x[/latex]

Find the derivative of [latex]g(x)=\frac{\cos x}{4x^2}[/latex].

Answer:

By applying the quotient rule, we have

[latex]g^{\prime}(x)=\frac{(−\sin x)4x^2-8x(\cos x)}{(4x^2)^2}[/latex].

Simplifying, we obtain

[latex]\begin{array}{ll}g^{\prime}(x) & =\frac{-4x^2 \sin x-8x \cos x}{16x^4} \\ & =\frac{−x \sin x-2 \cos x}{4x^3} \end{array}[/latex]

Find the derivative of [latex]f(x)=\frac{x}{\cos x}[/latex].

Answer:

[latex]\frac{\cos x+x \sin x}{\cos^2 x}[/latex]

Hint

Use the quotient rule.

An Application to Velocity

A particle moves along a coordinate axis in such a way that its position at time [latex]t[/latex] is given by [latex]s(t)=2 \sin t-t[/latex] for [latex]0\le t\le 2\pi[/latex]. At what times is the particle at rest?

Answer:

To determine when the particle is at rest, set [latex]s^{\prime}(t)=v(t)=0[/latex]. Begin by finding [latex]s^{\prime}(t)[/latex]. We obtain

[latex]s^{\prime}(t)=2 \cos t-1[/latex],

so we must solve

[latex]2 \cos t-1=0[/latex] for [latex]0\le t\le 2\pi[/latex].

The solutions to this equation are [latex]t=\frac{\pi}{3}[/latex] and [latex]t=\frac{5\pi}{3}[/latex]. Thus the particle is at rest at times [latex]t=\frac{\pi}{3}[/latex] and [latex]t=\frac{5\pi}{3}[/latex].

A particle moves along a coordinate axis. Its position at time [latex]t[/latex] is given by [latex]s(t)=\sqrt{3}t+2 \cos t[/latex] for [latex]0\le t\le 2\pi[/latex]. At what times is the particle at rest?

Answer:

[latex]t=\frac{\pi}{3}, \, t=\frac{2\pi}{3}[/latex]

Hint

Use the previous example as a guide.

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.

The Derivative of the Tangent Function

Find the derivative of [latex]f(x)= \tan x[/latex].

Answer:

Start by expressing [latex]\tan x[/latex] as the quotient of [latex]\sin x[/latex] and [latex]\cos x[/latex]:

[latex]f(x)= \tan x=\frac{\sin x}{\cos x}[/latex].

Now apply the quotient rule to obtain

[latex]f^{\prime}(x)=\frac{\cos x \cos x-(−\sin x)\sin x}{(\cos x)^2}[/latex].

Simplifying, we obtain

[latex]f^{\prime}(x)=\frac{\cos^2 x+\sin^2 x}{\cos^2 x}[/latex].

Recognizing that [latex]\cos^2 x+\sin^2 x=1[/latex], by the Pythagorean Identity, we now have

[latex]f^{\prime}(x)=\frac{1}{\cos^2 x}[/latex].

Finally, use the identity [latex]\sec x=\frac{1}{\cos x}[/latex] to obtain

[latex]f^{\prime}(x)=\sec^2 x[/latex].

Find the derivative of [latex]f(x)= \cot x[/latex].

Answer:

[latex]f^{\prime}(x)=−\csc^2 x[/latex]

Hint

Rewrite [latex]\cot x[/latex] as [latex]\frac{\cos x}{\sin x}[/latex] and use the quotient rule.

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

Derivatives of [latex]\tan x, \, \cot x, \, \sec x[/latex], and [latex]\csc x[/latex]

The derivatives of the remaining trigonometric functions are as follows:

[latex]\frac{d}{dx}(\tan x)=\sec^2 x[/latex]

[latex]\frac{d}{dx}(\cot x)=−\csc^2 x[/latex]

[latex]\frac{d}{dx}(\sec x)= \sec x \tan x[/latex]

[latex]\frac{d}{dx}(\csc x)=−\csc x \cot x[/latex]

Finding the Equation of a Tangent Line

Find the equation of a line tangent to the graph of [latex]f(x)= \cot x[/latex] at [latex]x=\frac{\pi}{4}[/latex].

Answer:

To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute

[latex]f(\frac{\pi}{4})= \cot \frac{\pi}{4}=1[/latex].

Thus the tangent line passes through the point [latex](\frac{\pi}{4},1)[/latex]. Next, find the slope by finding the derivative of [latex]f(x)= \cot x[/latex] and evaluating it at [latex]\frac{\pi}{4}[/latex]:

[latex]f^{\prime}(x)=−\csc^2 x[/latex] and [latex]f^{\prime}(\frac{\pi}{4})=−\csc^2 (\frac{\pi}{4})=-2[/latex].

Using the point-slope equation of the line, we obtain

[latex]y-1=-2(x-\frac{\pi}{4})[/latex]

or equivalently,

[latex]y=-2x+1+\frac{\pi}{2}[/latex].

Finding the Derivative of Trigonometric Functions

Find the derivative of [latex]f(x)= \csc x+x \tan x.[/latex]

Answer:

To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find

[latex]f^{\prime}(x)=\frac{d}{dx}(\csc x)+\frac{d}{dx}(x \tan x)[/latex].

In the first term, [latex]\frac{d}{dx}(\csc x)=−\csc x \cot x[/latex], and by applying the product rule to the second term we obtain

[latex]\frac{d}{dx}(x \tan x)=(1)(\tan x)+(\sec^2 x)(x)[/latex].

Therefore, we have

[latex]f^{\prime}(x)=−\csc x \cot x+ \tan x+x \sec^2 x[/latex].

Find the derivative of [latex]f(x)=2 \tan x-3 \cot x[/latex].

Answer:

[latex]f^{\prime}(x)=2 \sec^2 x+3 \csc^2 x[/latex]

Hint

Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.

Find the slope of the line tangent to the graph of [latex]f(x)= \tan x[/latex] at [latex]x=\frac{\pi}{6}[/latex].

Answer:

[latex]\frac{4}{3}[/latex]

Hint

Evaluate the derivative at [latex]x=\frac{\pi}{6}[/latex].

Key Concepts

  • We can find the derivatives of [latex]\sin x[/latex] and [latex]\cos x[/latex] by using the definition of derivative and the limit formulas found earlier. The results are

    [latex]\frac{d}{dx} \sin x= \cos x[/latex] and [latex]\frac{d}{dx} \cos x=−\sin x[/latex].

  • With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

For the following exercises, find [latex]\frac{dy}{dx}[/latex] for the given functions.

1.[latex]y=x^2- \sec x+1[/latex]

Answer: [latex]\frac{dy}{dx}=2x- \sec x \tan x[/latex]

2.[latex]y=3 \csc x+\frac{5}{x}[/latex]

3.[latex]y=x^2 \cot x[/latex]

Answer:

[latex]\frac{dy}{dx}=2x \cot x-x^2 \csc^2 x[/latex]

4.[latex]y=x-x^3 \sin x[/latex]

5.[latex]y=\frac{\sec x}{x}[/latex]

Answer:

[latex]\frac{dy}{dx}=\frac{x \sec x \tan x- \sec x}{x^2}[/latex]

6.[latex]y= \sin x \tan x[/latex]

7.[latex]y=(x+ \cos x)(1- \sin x)[/latex]

Answer:

[latex]\frac{dy}{dx}=(1- \sin x)(1- \sin x)- \cos x(x+ \cos x)[/latex]

8.[latex]y=\frac{\tan x}{1- \sec x}[/latex]

9.[latex]y=\frac{1- \cot x}{1+ \cot x}[/latex]

Answer:

[latex]\frac{dy}{dx}=\frac{2 \csc^2 x}{(1+ \cot x)^2}[/latex]

10.[latex]y= \cos x(1+ \csc x)[/latex]

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[/latex]. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

11. [T][latex]f(x)=−\sin x, \, x=0[/latex]

Answer:

[latex]y=−x[/latex]

Study Guide - Derivatives of Trigonometric Functions (4)

12. [T][latex]f(x)= \csc x, \, x=\frac{\pi}{2}[/latex]

13. [T][latex]f(x)=1+ \cos x, \, x=\frac{3\pi}{2}[/latex]

Answer:

[latex]y=x+\frac{2-3\pi}{2}[/latex]

Study Guide - Derivatives of Trigonometric Functions (5)

14. [T][latex]f(x)= \sec x, \, x=\frac{\pi}{4}[/latex]

15. [T][latex]f(x)=x^2- \tan x, \, x=0[/latex]

Answer:[latex-display]y=−x[/latex-display]Study Guide - Derivatives of Trigonometric Functions (6)

16. [T][latex]f(x)=5 \cot x, \, x=\frac{\pi}{4}[/latex]

For the following exercises, find [latex]\frac{d^2 y}{dx^2}[/latex] for the given functions.

17.[latex]y=x \sin x- \cos x[/latex]

Answer:

[latex]\frac{d^2 y}{dx^2} = 3 \cos x-x \sin x[/latex]

18.[latex]y= \sin x \cos x[/latex]

19.[latex]y=x-\frac{1}{2} \sin x[/latex]

Answer:

[latex]\frac{d^2 y}{dx^2} = \frac{1}{2} \sin x[/latex]

20.[latex]y=\frac{1}{x}+ \tan x[/latex]

21.[latex]y=2 \csc x[/latex]

Answer:

[latex]\frac{d^2 y}{dx^2} = \csc (x)(3 \csc^2 x-1+ \cot^2 x)[/latex]

22.[latex]y=\sec^2 x[/latex]

23.Find all [latex]x[/latex] values on the graph of [latex]f(x)=-3 \sin x \cos x[/latex] where the tangent line is horizontal.

Answer:

[latex]x = \frac{(2n+1)\pi}{4}[/latex], where [latex]n[/latex] is an integer

24.Find all [latex]x[/latex] values on the graph of [latex]f(x)=x-2 \cos x[/latex] for [latex]0<x<2\pi[/latex] where the tangent line has a slope of 2.

25.Let [latex]f(x)= \cot x[/latex]. Determine the point(s) on the graph of [latex]f[/latex] for [latex]0<x<2\pi[/latex] where the tangent line is parallel to the line [latex]y=-2x[/latex].

Answer:

[latex](\frac{\pi}{4},1), \, (\frac{3\pi}{4},-1)[/latex]

26. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 \cos t[/latex] where [latex]s[/latex] is measured in inches and [latex]t[/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[/latex] s.

27.Let the position of a swinging pendulum in simple harmonic motion be given by [latex]s(t)=a \cos t+b \sin t[/latex]. Find the constants [latex]a[/latex] and [latex]b[/latex] such that when the velocity is 3 cm/s, [latex]s=0[/latex] and [latex]t=0[/latex].

Answer:

[latex]a=0, \, b=3[/latex]

28.After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 \cos t[/latex] cm at [latex]t[/latex] seconds after the jump.

  1. Sketch one period of the position function for [latex]t\ge 0[/latex].
  2. Find the velocity function.
  3. Sketch one period of the velocity function for [latex]t\ge 0[/latex].
  4. Determine the times when the velocity is 0 over one period.
  5. Find the acceleration function.
  6. Sketch one period of the acceleration function for [latex]t\ge 0[/latex].

29.The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 \sin x[/latex] where [latex]y[/latex] is the number of hamburgers sold and [latex]x[/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y^{\prime}[/latex] and determine the intervals where the number of burgers being sold is increasing.

Answer:

[latex]y^{\prime}=5 \cos (x)[/latex], increasing on [latex](0,\frac{\pi}{2}), \, (\frac{3\pi}{2},\frac{5\pi}{2})[/latex], and [latex](\frac{7\pi}{2},12)[/latex]

30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 \cos t[/latex], where [latex]t[/latex] is the number of months since January. Find [latex]y^{\prime}[/latex] 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.

31.[latex]\frac{d}{dx}(\cot x)=−\csc^2 x[/latex]

32.[latex]\frac{d}{dx}(\sec x)= \sec x \tan x[/latex]

33.[latex]\frac{d}{dx}(\csc x)=−\csc x \cot x[/latex]

34.Use the definition of derivative and the identity

[latex]\cos (x+h)= \cos x \cos h- \sin x \sin h[/latex] to prove that [latex]\frac{d}{dx}(\cos x)=−\sin x[/latex].

For the following exercises, find the requested higher-order derivative for the given functions.

35.[latex]\frac{d^3 y}{dx^3}[/latex] of [latex]y=3 \cos x[/latex]

Answer: [latex]\frac{d^3 y}{dx^3} = 3 \sin x[/latex]

36.[latex]\frac{d^2 y}{dx^2}[/latex] of [latex]y=3 \sin x+x^2 \cos x[/latex]

37.[latex]\frac{d^4 y}{dx^4}[/latex] of [latex]y=5 \cos x[/latex]

Answer:

[latex]\frac{d^4 y}{dx^4} = 5 \cos x[/latex]

38.[latex]\frac{d^2 y}{dx^2}[/latex] of [latex]y= \sec x+ \cot x[/latex]

39.[latex]\frac{d^3 y}{dx^3}[/latex] of [latex]y=x^{10}- \sec x[/latex]

Answer:

[latex]\frac{d^3 y}{dx^3} = 720x^7-5 \tan (x) \sec^3 (x)- \tan^3 (x) \sec (x)[/latex]

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I have a solid understanding of calculus, particularly in the context of trigonometric functions and their derivatives. The provided article discusses the exploration of the derivative for the sine function and extends to various trigonometric functions. I'll provide information related to the concepts used in the article.

  1. Derivative of Sine Function:

    • The derivative of the sine function, denoted as [latex]\frac{d}{dx}(\sin x)[/latex], is given by [latex]\cos x[/latex].
    • The article uses the definition of the derivative and trigonometric identities to prove this result.
  2. Derivative of Cosine Function:

    • The derivative of the cosine function, denoted as [latex]\frac{d}{dx}(\cos x)[/latex], is [latex]-\sin x[/latex].
    • The proofs for both [latex]\frac{d}{dx}(\sin x)[/latex] and [latex]\frac{d}{dx}(\cos x)[/latex] use similar techniques involving trigonometric limits and identities.
  3. Derivatives of Other Trigonometric Functions:

    • The article provides derivative formulas for tangent, cotangent, secant, and cosecant functions using the quotient rule and other trigonometric identities.
    • Examples are given for finding the derivatives of these functions.
  4. Applications of Derivatives:

    • Tangent lines to trigonometric functions are discussed, including finding equations for tangent lines at specific points.
    • Applications to physics, such as simple harmonic motion and oscillations, are explored.
  5. Higher-Order Derivatives:

    • The article briefly touches on finding higher-order derivatives of trigonometric functions.
    • Examples involve finding second and third derivatives for specific trigonometric expressions.
  6. Velocity and Acceleration:

    • There are applications to velocity, where the article discusses finding the rate of oscillation for a spring and determining when a particle is at rest.
  7. Equations of Tangent Lines:

    • The article provides examples of finding equations for tangent lines to trigonometric functions at specific points.
  8. Graphical Interpretation:

    • Graphs of trigonometric functions and their derivatives are used to illustrate concepts such as points of horizontal tangency and slopes of tangent lines.

I can provide further details or answer specific questions related to these concepts.

Study Guide - Derivatives of Trigonometric Functions (2024)
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