Derivatives of Sine and Cosine | MIT 18.01SC Single Variable Calculus, Fall 2010

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7 Jan 201108:11
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TLDRIn this recitation, the professor revisits the topic of derivatives, specifically focusing on trigonometric functions. The example given is a function h(x) involving sine and cosine terms. The challenge is to find the values of x where the derivative of h(x) equals zero. The solution process involves applying the sum and constant multiple rules of differentiation, leading to a tangent equation. The solutions are found to be x = Ο€/6 plus multiples of Ο€, illustrating the periodic nature of the tangent function. An alternative approach using trigonometric identities is also discussed, providing a deeper understanding of the function's behavior and its horizontal tangent lines.

Takeaways
  • πŸ“š The lesson is a continuation of a previous lecture on derivatives, specifically focusing on trigonometric functions like sine and cosine.
  • πŸ” The example function provided is \( h(x) = \sin(x) + \sqrt{3}\cos(x) \), and the task is to find the values of \( x \) where the derivative of \( h(x) \) is zero.
  • πŸ“ The derivative of \( h(x) \) is calculated by applying the rules of differentiation to the sum of sine and cosine functions, resulting in \( h'(x) = \cos(x) - \sqrt{3}\sin(x) \).
  • πŸ”„ The process involves using the sum rule of derivatives and the derivatives of sine and cosine functions, which are \( \cos(x) \) and \( -\sin(x) \), respectively.
  • 🌐 The equation \( h'(x) = 0 \) is solved by manipulating the terms to isolate \( \tan(x) \), leading to the equation \( \tan(x) = \frac{1}{\sqrt{3}} \).
  • πŸ“ Recognizing the tangent function's periodicity, the solutions for \( x \) are given by \( x = \frac{\pi}{6} + k\pi \), where \( k \) is any integer.
  • πŸ“ˆ An alternative approach involves rewriting the original function using trigonometric identities to resemble the sine of a sum formula, leading to \( h(x) = 2\sin(x + \frac{\pi}{3}) \).
  • πŸ“‰ The horizontal tangent lines on the graph of \( h(x) \) correspond to the points where the derivative is zero, which can be found by understanding the graph's behavior.
  • πŸ“š The lecture also explains how to find the derivative of a function without calculus, by using the properties of the sine function and its transformations.
  • πŸ“Š The graph of \( h(x) \) is derived from shifting and scaling the graph of \( y = \sin(x) \), and the points of interest are where the graph has horizontal tangents.
  • πŸ”‘ The key takeaway is the application of calculus, algebra, and trigonometry to solve for the values of \( x \) that satisfy the condition \( h'(x) = 0 \) in the context of a trigonometric function.
Q & A

  • What was the main topic of the lecture?

    -The main topic of the lecture was finding the derivatives of trigonometric functions, specifically focusing on the sine and cosine functions.

  • What is the given function h(x) in the script?

    -The given function h(x) is h(x) = sine(x) + √3 * cosine(x).

  • What is the task the professor assigns to the students?

    -The task is to find the values of x for which the derivative of h(x) is equal to 0.

  • What is the derivative of the sum of two functions?

    -The derivative of the sum of two functions is the sum of the derivatives of the individual functions.

  • What are the derivatives of sine(x) and cosine(x)?

    -The derivative of sine(x) is cosine(x), and the derivative of cosine(x) is -sine(x).

  • What is the formula for the derivative of h(x)?

    -The derivative of h(x), denoted as h'(x), is cosine(x) - √3 * sine(x).

  • How does the professor suggest solving the equation h'(x) = 0?

    -The professor suggests isolating x by adding √3 * sine(x) to one side and dividing by cosine(x), which leads to the equation tan(x) = 1/√3.

  • What is the simplest solution for x when tan(x) = 1/√3?

    -The simplest solution for x is x = Ο€/6 (or 30 degrees).

  • What is the period of the tangent function?

    -The period of the tangent function is Ο€.

  • How can the solutions for x be expressed in a general form?

    -The solutions for x can be expressed in the general form of Ο€/6 + nΟ€, where n is an arbitrary integer.

  • What alternative approach does the professor introduce to solve the problem?

    -The alternative approach involves using trigonometric identities to rewrite h(x) as 2 * sine(x + Ο€/3) and then analyzing the graph for points where the derivative is 0.

  • What is the significance of the angle addition formula in the alternative approach?

    -The angle addition formula is used to rewrite h(x) in a form that makes it easier to identify the points where the derivative is 0 by visual inspection of the graph.

  • How does the professor describe the graph of y = 2 * sine(x + Ο€/3)?

    -The graph is described as a sine wave that has been shifted left by Ο€/3 and scaled up by a factor of 2.

  • What are the critical points of interest on the graph of y = 2 * sine(x + Ο€/3)?

    -The critical points of interest are the points where there is a horizontal tangent line, indicating where the derivative is 0.

Outlines
00:00
πŸ“š Derivative of Trigonometric Functions Practice

The professor begins by welcoming students to a recitation session focused on the derivatives of trigonometric functions, specifically sine and cosine. An example function, h(x) = sin(x) + √3cos(x), is introduced to illustrate the application of these derivatives. Students are encouraged to pause the video and attempt to find the values of x where the derivative of h(x) equals zero. The professor then guides through the process of finding the derivative, h'(x) = cos(x) - √3sin(x), and setting it equal to zero to solve for x. The approach involves algebraic manipulation to express the equation in terms of tan(x) = 1/√3, leading to the solution x = Ο€/6. The periodic nature of the tangent function is highlighted, indicating that the solutions are of the form x = Ο€/6 + nΟ€, where n is an integer.

05:01
πŸ” Alternative Solution Using Trigonometric Identities

The second paragraph presents an alternative method to solve the derivative problem using trigonometric identities. The function h(x) is rewritten by multiplying and dividing by 2 to resemble the sine angle addition formula. This transformation allows the function to be expressed as 2sin(x + Ο€/3), leveraging known trigonometric identities. The derivative of this new form is implicitly zero at the points where the sine function has a horizontal tangent, which are x = Ο€/6 and x = 7Ο€/6, corresponding to the peaks and troughs of the sine wave shifted by Ο€/3. The professor illustrates this with a schematic graph, explaining the shift and scaling effects on the sine function, and concludes by identifying the specific x values where the derivative is zero.

Mindmap
Keywords
πŸ’‘Derivative
In calculus, the derivative of a function measures the rate at which the function's value changes with respect to changes in its input variable. It is a fundamental concept used to analyze the behavior of functions. In the video, the professor is focusing on finding the derivative of a composite function involving trigonometric functions, specifically sine and cosine, to determine the points where the rate of change is zero.
πŸ’‘Trigonometric functions
Trigonometric functions are mathematical functions relating the angles of a triangle to the lengths of its sides. In the script, sine and cosine are mentioned as the primary trigonometric functions involved in the function h(x). These functions are essential in the context of the video as they are the building blocks of the function whose derivative is being analyzed.
πŸ’‘Sine function
The sine function is a trigonometric function that relates the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. In the video, the sine function is one of the components of the main function h(x), and its derivative is used to find the critical points where the derivative of h(x) equals zero.
πŸ’‘Cosine function
The cosine function is another trigonometric function, similar to sine but relating to the adjacent side of a right triangle instead. In the script, the cosine function is multiplied by the square root of 3 and added to the sine function to form h(x), and its derivative is crucial in solving for the values of x where h'(x) is zero.
πŸ’‘Square root of 3
The square root of 3 is a mathematical constant that appears in the script as a coefficient for the cosine function within the function h(x). It is used to adjust the amplitude of the cosine wave and is part of the process to find the derivative of h(x) and solve for x values where the derivative is zero.
πŸ’‘Critical points
Critical points are the points on the graph of a function where the derivative is zero or undefined. In the context of the video, the professor is looking for critical points of the function h(x) by setting its derivative equal to zero and solving for x, which will help in understanding the function's behavior.
πŸ’‘Tangent function
The tangent function is a trigonometric function defined as the ratio of the sine to the cosine of an angle. In the script, the professor uses the tangent function to rewrite the equation derived from setting the derivative of h(x) to zero, which simplifies to tan(x) = 1/√3, helping to find the critical points.
πŸ’‘Arc tangent function
The arc tangent function, often abbreviated as atan or tan^(-1), is the inverse function of the tangent. It is used to find the angle whose tangent is a given number. In the video, the professor suggests using the arc tangent to find the values of x that satisfy the equation tan(x) = 1/√3.
πŸ’‘Periodic function
A periodic function is a function that repeats its values at regular intervals or periods. In the script, the professor mentions that the tangent function is periodic with a period of Ο€, which is important for understanding the infinite number of solutions to the equation tan(x) = 1/√3.
πŸ’‘Trigonometric identities
Trigonometric identities are equations that hold true for all values of the variables that they contain. In the video, the professor uses the sum-to-product trigonometric identities to rewrite the function h(x) in a form that resembles the sine of a sum, which simplifies the process of finding the derivative and solving for critical points.
πŸ’‘Angle addition formula
The angle addition formula is a trigonometric identity that allows you to express the sine or cosine of the sum of two angles in terms of the sines and cosines of the individual angles. In the script, the professor applies the sine angle addition formula in reverse to rewrite h(x) as 2 sine(x + Ο€/3), providing an alternative method to find the critical points.
Highlights

Introduction to the topic of finding derivatives of trigonometric functions, specifically sine and cosine.

Presentation of the function h(x) = sin(x) + √3cos(x) and the task to find values of x where its derivative is zero.

Encouragement for students to pause the video and attempt the problem independently.

Explanation of the derivative of a sum of functions and the application of derivative rules for sine and cosine.

Derivation of h'(x) = cos(x) - √3sin(x) using the constant multiple rule.

Setting up the equation to find when h'(x) = 0 leads to cos(x) - √3sin(x) = 0.

Preferred method of solving the equation by adding √3sin(x) to one side and dividing by cos(x).

Transformation of the equation to tan(x) = 1/√3 using trigonometric identities.

Identification of the special trigonometric angle x = Ο€/6 that satisfies the equation.

Discussion on the periodicity of the tangent function and the infinite solutions for x.

General formula for the solutions: x = Ο€/6 + nΟ€, where n is an integer.

Introduction of an alternative approach using trigonometric identities to simplify h(x).

Rewriting h(x) using angle addition formula for sine and identifying it as 2sin(x + Ο€/3).

Graphical representation of the function and its horizontal tangent lines.

Identification of additional solutions for x based on the shifted sine function.

Explanation of the minimum point of the function at x = 7Ο€/6 and its significance.

Summary of the two methods used to solve for the derivative being zero and their outcomes.

Transcripts

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TrigonometryDerivativesSine FunctionCosine FunctionCalculusMathematicsProblem SolvingEducational ContentRecitation ClassTrig Identities