Find All Solutions To The Equation -sin(x) = Cos^2(x) - 1 In The Interval [0, 2π].

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Introduction

In this article, we delve into the process of finding all solutions for the trigonometric equation -sin(x) = cos²(x) - 1 within the interval [0, 2π]. Trigonometric equations, which involve trigonometric functions like sine, cosine, and tangent, play a pivotal role in various fields including physics, engineering, and mathematics. Solving these equations often requires a blend of trigonometric identities, algebraic manipulation, and a keen understanding of the unit circle. This exploration will not only enhance your problem-solving skills but also deepen your comprehension of trigonometric functions and their applications.

Understanding the Basics of Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions of an unknown angle. The solutions to these equations are the angles that satisfy the given equation. Unlike algebraic equations that typically have a finite number of solutions, trigonometric equations can have infinitely many solutions due to the periodic nature of trigonometric functions. For instance, the sine function, sin(x), repeats its values every 2π radians, meaning that if x is a solution to a trigonometric equation, then x + 2πk is also a solution for any integer k. However, when solving trigonometric equations, we are often interested in finding solutions within a specific interval, such as [0, 2π], which represents one complete cycle of the trigonometric functions.

To effectively solve trigonometric equations, it is essential to have a firm grasp of trigonometric identities. These identities are equations that are true for all values of the variables involved. Some fundamental trigonometric identities include:

  1. Pythagorean Identity: sin²(x) + cos²(x) = 1
  2. Reciprocal Identities: csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x)
  3. Quotient Identities: tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)
  4. Angle Sum and Difference Identities:
    • sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
    • cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
    • tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))
  5. Double Angle Identities:
    • sin(2x) = 2sin(x)cos(x)
    • cos(2x) = cos²(x) - sin²(x) = 1 - 2sin²(x) = 2cos²(x) - 1
    • tan(2x) = 2tan(x) / (1 - tan²(x))
  6. Half Angle Identities:
    • sin(x/2) = ±√((1 - cos(x)) / 2)
    • cos(x/2) = ±√((1 + cos(x)) / 2)
    • tan(x/2) = sin(x) / (1 + cos(x)) = (1 - cos(x)) / sin(x)

These identities allow us to rewrite trigonometric expressions in different forms, which can simplify equations and make them easier to solve. For example, the Pythagorean identity sin²(x) + cos²(x) = 1 is frequently used to express cos²(x) in terms of sin²(x) or vice versa, which is a crucial step in solving our given equation.

Solving the Equation -sin(x) = cos²(x) - 1

Now, let's address the equation at hand: -sin(x) = cos²(x) - 1. Our goal is to find all values of x within the interval [0, 2π] that satisfy this equation. To begin, we can use the Pythagorean identity to rewrite the equation in terms of a single trigonometric function. Recall that sin²(x) + cos²(x) = 1. Rearranging this identity, we get cos²(x) = 1 - sin²(x). Substituting this into our equation gives:

-sin(x) = (1 - sin²(x)) - 1

Simplifying the equation, we have:

-sin(x) = -sin²(x)

Now, we can move all terms to one side of the equation to set it equal to zero:

sin²(x) - sin(x) = 0

This equation looks like a quadratic equation in terms of sin(x). To make this more apparent, let's substitute y = sin(x). The equation then becomes:

y² - y = 0

This is a simple quadratic equation that can be factored:

y(y - 1) = 0

This gives us two possible solutions for y:

y = 0 or y = 1

Now, we substitute back sin(x) for y to find the values of x:

sin(x) = 0 or sin(x) = 1

Finding the Solutions for sin(x) = 0

To solve sin(x) = 0 in the interval [0, 2π], we need to identify the angles where the sine function is equal to zero. Recall that the sine function represents the y-coordinate on the unit circle. The y-coordinate is zero at two points on the unit circle: the points corresponding to 0 radians and π radians. Additionally, since we are considering the interval [0, 2π], we must also include the angle 2π, where the sine function is also zero.

Therefore, the solutions for sin(x) = 0 in the interval [0, 2π] are:

  • x = 0
  • x = π
  • x = 2π

These angles represent the points where the unit circle intersects the x-axis.

Finding the Solutions for sin(x) = 1

Next, we need to solve sin(x) = 1 in the interval [0, 2π]. The sine function is equal to 1 when the y-coordinate on the unit circle is 1. This occurs at only one point in the interval [0, 2π], which corresponds to the angle π/2 radians.

Therefore, the solution for sin(x) = 1 in the interval [0, 2π] is:

  • x = π/2

This angle represents the point where the unit circle intersects the positive y-axis.

Combining the Solutions

Now that we have found the solutions for both sin(x) = 0 and sin(x) = 1, we can combine them to obtain the complete set of solutions for the original equation -sin(x) = cos²(x) - 1 in the interval [0, 2π]. The solutions are:

  • x = 0
  • x = π/2
  • x = π
  • x = 2π

These four angles are the only values within the interval [0, 2π] that satisfy the given equation. It is always a good practice to verify these solutions by plugging them back into the original equation to ensure they are correct.

Verification of Solutions

To verify our solutions, we will substitute each value of x back into the original equation, -sin(x) = cos²(x) - 1:

  1. For x = 0:
    • -sin(0) = 0
    • cos²(0) - 1 = 1² - 1 = 0
    • The equation holds true.
  2. For x = π/2:
    • -sin(π/2) = -1
    • cos²(π/2) - 1 = 0² - 1 = -1
    • The equation holds true.
  3. For x = π:
    • -sin(π) = 0
    • cos²(π) - 1 = (-1)² - 1 = 0
    • The equation holds true.
  4. For x = 2π:
    • -sin(2π) = 0
    • cos²(2π) - 1 = 1² - 1 = 0
    • The equation holds true.

Since all four solutions satisfy the original equation, we can confidently conclude that they are correct.

Conclusion

In this comprehensive guide, we have successfully found all solutions for the trigonometric equation -sin(x) = cos²(x) - 1 within the interval [0, 2π]. By leveraging the Pythagorean identity, factoring techniques, and a solid understanding of the unit circle, we were able to simplify the equation and identify the solutions. The solutions are x = 0, x = π/2, x = π, and x = 2π. We also verified these solutions by substituting them back into the original equation, ensuring their correctness.

Mastering the art of solving trigonometric equations is a valuable skill that has applications in various scientific and engineering disciplines. By understanding the fundamental concepts and techniques, you can tackle a wide range of trigonometric problems with confidence.