An Interesting Trigonometric Equation | Viewer Suggested

SyberMath
11 Mar 202407:43
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, the presenter tackles a trigonometric problem involving the sign of cosine and sine functions, as suggested by viewer Clement HJ. The problem is to solve for x when the sign of cosine of x equals the cosine of the sign of x. The presenter explains that there are no real solutions due to the constraints of the cosine function and its maximum value. However, the video also explores complex solutions using the double angle formula and the quadratic formula, leading to non-real solutions expressed in terms of Euler's formula.

Takeaways
  • πŸ“Œ The video discusses a trigonometric problem involving the sign of cosine of x and its relationship with cosine of sin of x.
  • πŸ” The problem was suggested by a viewer named Clement HJ, and the video aims to solve it step by step.
  • πŸ€” The initial approach is to transform the problem into a form where both sides are the same trigonometric function to simplify the equation.
  • 🌐 The cosine function is rewritten using the identity that cosine of an angle can be expressed as sine of its complement.
  • πŸ“Š The video uses the properties of complementary angles and the fact that if two angles are supplementary, their signs are equal.
  • 🚫 It is determined that there are no real solutions to the equation based on the upper bound of the sum of cosine and sine functions.
  • πŸŒ€ The video then explores the possibility of complex or non-real solutions by squaring both sides of the equation.
  • πŸ“ The double angle formula for sine is used to further simplify the equation and find potential solutions.
  • πŸ”’ The video introduces a quadratic equation to solve for complex solutions and uses the substitution method to find the value of T.
  • πŸŽ“ The problem-solving process involves using the Euler formula to handle complex solutions and finding the value of e to the i Alpha.
  • πŸ“ˆ The video concludes that the solutions to the equation are non-real, as the value derived from the quadratic equation exceeds the maximum value for sine.
Q & A

  • What is the trigonometric problem presented in the video?

    -The problem is to solve the equation where the sign of cosine of x is equal to the cosine of the sine of x.

  • How does the video suggest to transform the given equation?

    -The video suggests transforming the equation by replacing the cosine with the sign of something, using the identity that cosine of alpha can always be written as the sine of Pi/2 minus alpha.

  • What is the significance of considering the complementary angles in this problem?

    -Considering complementary angles is significant because it allows the transformation of the equation into a form where sine and cosine functions are of the same angle, making it easier to solve.

  • Why does the video mention that there are no real solutions to the equation?

    -The video mentions that there are no real solutions because the upper bound for cosine x + sin x is less than Pi/2, which is greater than the maximum value for the function, indicating that the equation does not hold for real values of x.

  • How does the video approach the second branch of the problem?

    -The video approaches the second branch by considering one angle to be Pi minus the other, leading to an equation where cosine x - sin x equals Pi/2 plus 2n*pi, and then proceeds to find complex solutions.

  • What is the role of the double angle formula in solving for complex solutions?

    -The double angle formula is used to express the equation in terms of sin of 2x, which is then used to find the complex solutions by setting up a quadratic equation in terms of e to the i Alpha.

  • How does the video use Euler's formula to find the complex solutions?

    -The video uses Euler's formula to express sin Alpha in terms of e to the i Alpha, which is then substituted into the quadratic equation to solve for the complex solutions.

  • What is the final form of the quadratic equation derived in the video?

    -The final form of the quadratic equation derived in the video is T^2 - T - (Pi^2 - 4)/2i = 0.

  • What is the main conclusion of the video regarding the original trigonometric problem?

    -The main conclusion is that the original trigonometric problem does not have any real solutions, but it does have non-real or complex solutions.

  • How does the video utilize the concept of supplementary angles in the problem?

    -The video utilizes the concept of supplementary angles by considering the case where the cosine x and sine x are supplementary, leading to the conclusion that the solutions are non-real since their sum exceeds the maximum value of the function.

  • What is the significance of the upper bound of cosine x + sin x in the problem?

    -The significance of the upper bound is that it helps determine the possible range of values for the function, which in this case indicates that there are no real solutions because the upper bound is less than Pi/2.

Outlines
00:00
πŸ“š Trigonometric Problem Solving

The video begins with the introduction of a trigonometric problem involving the sign of cosine of x and cosine of sin of x. The problem was suggested by a viewer named Clement HJ. The aim is to solve this problem by transforming the trigonometric expressions into a form where they can be compared. The video explains how to use the complementary angle identity to rewrite the sine of cosine of x as sin(Ο€/2 - x) and then proceeds to analyze the possible solutions by considering the sum of cosine x and sine x. It is concluded that there are no real solutions to the equation because the maximum value of the sum exceeds the boundary for real numbers.

05:02
πŸ” Complex Solutions for Trigonometric Equations

The second paragraph delves into finding complex solutions for the given trigonometric equation. It explains the process of squaring both sides of the equation to simplify it and then using the double angle formula for sine. The video outlines the steps to solve for the complex solutions using the quadratic formula and the substitution method. It emphasizes that the solutions are non-real because the value obtained from the equation exceeds the boundary for real numbers. The video also introduces the concept of Euler's formula to express the complex solutions in the form of e to the power of an imaginary number.

Mindmap
Keywords
πŸ’‘Trigonometric problem
A mathematical problem involving the relationships between the angles and sides of triangles, typically using functions such as sine, cosine, and tangent. In the video, the problem involves the equation 'sign of cosine of x equals cosine of sin of X', which is an unconventional trigonometric equation proposed by a viewer.
πŸ’‘Sign function
A mathematical function that returns the sign (positive, negative, or zero) of a given real number. In the context of the video, it is used to describe the sign of the cosine function, which is part of the complex trigonometric equation being solved.
πŸ’‘Cosine
One of the six trigonometric functions, often denoted by 'cos', it relates the angle of a right triangle to the ratio of the length of its adjacent side to the hypotenuse. In the video, the cosine function is central to the problem being solved, with the sign of its value being a key part of the equation.
πŸ’‘Sine
Another of the six trigonometric functions, denoted by 'sin', it represents the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. In the video, the sine function is used in conjunction with the cosine function to form the complex equation.
πŸ’‘Complementary angles
Two angles that add up to 90 degrees. In the context of the video, the concept is used to express the cosine function in terms of the sine function by recognizing that the cosine of an angle is equal to the sine of its complementary angle.
πŸ’‘Supplementary angles
Two angles that add up to 180 degrees. In the video, the concept is used to discuss the potential solutions of the trigonometric equation, noting that if two angles are supplementary, their signs are also equal.
πŸ’‘Upper bound
A value that is greater than or equal to the function's maximum value. In the video, the upper bound concept is used to determine that 'cosine X plus sin x' must be less than or equal to the square root of 2, which helps in concluding that there are no real solutions to the equation.
πŸ’‘Complex solutions
Solutions to mathematical problems that involve complex numbers, which are numbers with real and imaginary parts. In the video, after concluding that there are no real solutions, the focus shifts to finding complex solutions for the trigonometric equation.
πŸ’‘Double angle formula
A trigonometric identity that states that the sine of twice an angle is equal to twice the sine of the angle times the cosine of the angle. In the video, this formula is used to simplify the equation and find the complex solutions.
πŸ’‘Quadratic equation
A polynomial equation of degree two, in the form of ax^2 + bx + c = 0. In the video, the quadratic equation is derived from the process of solving for the complex solutions of the trigonometric problem.
πŸ’‘Graph
A visual representation of data or functions, showing the relationship between variables. In the video, the mention of a graph suggests that visual analysis could provide further insight into the interesting aspects of the trigonometric problem being solved.
Highlights

The video discusses solving a trigonometric problem involving the sign of cosine of x and cosine of sin of x.

The problem was suggested by a viewer named Clement HJ.

The goal is to express both trigonometric expressions in the same form to solve the equation.

Cosine of an angle can be written as sine of its complementary angle (Pi/2 - Alpha).

The video explores two cases for the given trigonometric equation.

The first case involves the sum of cosine x and sine x being less than or equal to the square root of 2.

The video concludes that there are no real solutions for the first case due to the upper bound of the function.

The second case considers cosine x as Pi minus the other angle, leading to a similar inequality issue.

The video explains that the solutions for the second case are non-real, using the double angle formula for sine.

The Euler formula is introduced to solve for complex solutions.

The video simplifies the complex solutions by setting n to zero and focusing on the case where 2x equals Alpha.

A quadratic equation is derived to solve for the complex solutions.

The video emphasizes the importance of recognizing the boundaries of trigonometric functions to determine the nature of solutions.

The process of isolating and solving for the complex solutions is detailed through algebraic manipulation.

The video provides a comprehensive walkthrough of solving trigonometric equations, including handling complex solutions.

The video concludes that the trigonometric problem has no real solutions but does have non-real solutions.

The video's approach to solving the problem is methodical and educational, suitable for viewers with a background in trigonometry.

The video demonstrates the practical application of trigonometric identities and the Euler formula in solving complex problems.

Transcripts

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