Solve The Equation $4 \sin^2 X = 5 + 4 \cos X$ In The Interval $[0, 2\pi)$. Express The Answer In Radians In Terms Of $\pi$.

In the realm of mathematics, trigonometric equations often present a fascinating challenge. These equations involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. Solving them requires a blend of algebraic manipulation, trigonometric identities, and a solid understanding of the unit circle. This article delves into the process of finding solutions for trigonometric equations within a specified interval, focusing on the equation $4 \sin^2 x = 5 + 4 \cos x$ in the interval $[0, 2\pi)$. We will explore the steps involved, from transforming the equation into a solvable form to identifying all solutions within the given range. Understanding these techniques is crucial for anyone studying trigonometry, calculus, or related fields. So, let's embark on this mathematical journey and unravel the solutions to this intriguing equation.

Understanding Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions of an unknown angle. The solutions to these equations are the angles that satisfy the equation. Unlike algebraic equations, trigonometric equations often have infinitely many solutions due to the periodic nature of trigonometric functions. For instance, the sine function repeats its values every $2\pi$ radians. Therefore, if $x$ is a solution to $\sin x = 0.5$, then $x + 2\pi$, $x + 4\pi$, and so on, are also solutions. This periodicity necessitates specifying an interval to find a finite set of solutions. The interval $[0, 2\pi)$ is commonly used, as it represents one full cycle around the unit circle. Within this interval, we can pinpoint all the unique angles that satisfy the given equation. The process of solving trigonometric equations often involves using trigonometric identities to simplify the equation, isolating the trigonometric function, and then finding the angles that correspond to the resulting value. It’s a blend of algebraic manipulation and trigonometric knowledge that makes this a rewarding mathematical endeavor. By mastering these techniques, we can unlock the solutions to a wide range of trigonometric problems.

Solving the Equation $4 \sin^2 x = 5 + 4 \cos x$

To solve the equation $4 \sin^2 x = 5 + 4 \cos x$, we need to employ a strategic approach that leverages trigonometric identities and algebraic manipulation. Our primary goal is to transform the equation into a form that is easier to solve. The presence of both sine and cosine functions suggests using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to express the equation in terms of a single trigonometric function. Substituting $\sin^2 x = 1 - \cos^2 x$ into the equation, we get: $4(1 - \cos^2 x) = 5 + 4 \cos x$. Expanding and rearranging the terms, we obtain a quadratic equation in terms of $\cos x$: $4 - 4 \cos^2 x = 5 + 4 \cos x$ which simplifies to $4 \cos^2 x + 4 \cos x + 1 = 0$. This quadratic equation can be factored as $(2 \cos x + 1)^2 = 0$. Setting the factor equal to zero gives us $2 \cos x + 1 = 0$, which leads to $\cos x = -\frac{1}{2}$. Now, we need to find the angles $x$ in the interval $[0, 2\pi)$ that satisfy this condition. Recognizing the cosine values of special angles is crucial here. We know that cosine is negative in the second and third quadrants. The reference angle for $\cos x = \frac{1}{2}$ is $\frac{\pi}{3}$. Therefore, the solutions in the interval $[0, 2\pi)$ are $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$ and $x = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$. Thus, the solutions to the equation are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$. This step-by-step process demonstrates the power of combining trigonometric identities and algebraic techniques to solve complex equations.

Step 1: Transforming the Equation

The initial step in solving the equation $4 \sin^2 x = 5 + 4 \cos x$ is to transform it into a more manageable form. This transformation hinges on a fundamental trigonometric identity: the Pythagorean identity. The Pythagorean identity, ${\sin^2 x + \cos^2 x = 1}$, provides a crucial link between the sine and cosine functions. By rearranging this identity, we can express ${\sin^2 x}$ in terms of ${\cos^2 x}$, or vice versa. In this particular equation, the presence of both ${\sin^2 x}$ and ${\cos x}$ terms suggests that expressing ${\sin^2 x}$ in terms of ${\cos^2 x}$ will simplify the equation. Substituting ${\sin^2 x = 1 - \cos^2 x}$ into the original equation yields: $4(1 - \cos^2 x) = 5 + 4 \cos x$. This substitution is a critical step because it reduces the number of trigonometric functions present in the equation, making it easier to manipulate algebraically. The transformed equation now involves only the cosine function, paving the way for further simplification and solution. This technique of using trigonometric identities to simplify equations is a cornerstone of solving trigonometric problems. It allows us to convert complex expressions into simpler, more solvable forms. The ability to recognize and apply these identities is a key skill for anyone working with trigonometric equations.

Step 2: Forming a Quadratic Equation

After substituting the Pythagorean identity, the equation $4(1 - \cos^2 x) = 5 + 4 \cos x$ needs further simplification. Expanding the left side of the equation gives us $4 - 4 \cos^2 x = 5 + 4 \cos x$. The next step is to rearrange the terms to form a quadratic equation. To do this, we move all terms to one side of the equation, setting the other side equal to zero. Adding $4 \cos^2 x$ and subtracting 4 from both sides, we get: $0 = 4 \cos^2 x + 4 \cos x + 1$. This equation is now in the standard quadratic form, which can be written as $a\cos^2 x + b \cos x + c = 0$, where $a = 4$, $b = 4$, and $c = 1$. Recognizing this quadratic form is crucial because it allows us to apply techniques for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation can be easily factored. The ability to transform a trigonometric equation into a quadratic equation is a powerful technique. It allows us to leverage our knowledge of algebra to solve trigonometric problems. By recognizing the underlying algebraic structure, we can apply familiar methods to find the solutions.

Step 3: Solving the Quadratic Equation

With the equation in quadratic form, $4 \cos^2 x + 4 \cos x + 1 = 0$, the next step is to solve for ${\cos x}$. This quadratic equation is particularly amenable to factoring. We look for two binomials that, when multiplied, give us the quadratic expression. In this case, the equation factors neatly as $(2 \cos x + 1)(2 \cos x + 1) = 0$, which can also be written as $(2 \cos x + 1)^2 = 0$. This factorization simplifies the problem significantly. Now, we set the factor equal to zero: $2 \cos x + 1 = 0$. Solving for ${\cos x}$, we subtract 1 from both sides and then divide by 2, resulting in ${\cos x = -\frac{1}{2}}$. This equation tells us that we are looking for angles $x$ whose cosine is equal to $-\frac{1}{2}$. Solving this equation is a critical step in finding the solutions to the original trigonometric equation. It reduces the problem to finding the angles that satisfy a specific cosine value. The ability to solve quadratic equations is a fundamental skill in mathematics, and its application here highlights the interplay between algebra and trigonometry. By factoring the quadratic equation, we have simplified the problem and made it easier to find the values of ${\cos x}$ that satisfy the original equation.

Step 4: Finding the Solutions in the Interval [0, 2π)

Now that we have ${\cos x = -\frac{1}{2}}$, the final step is to find the angles $x$ in the interval $[0, 2\pi)$ that satisfy this condition. To do this, we need to consider the unit circle and the quadrants where cosine is negative. Cosine is negative in the second and third quadrants. The reference angle, which is the angle in the first quadrant with the same cosine value (ignoring the sign), can be found by considering the angle whose cosine is ${\frac{1}{2}}$. This reference angle is ${\frac{\pi}{3}}$. In the second quadrant, the angle $x$ is given by ${\pi - \frac{\pi}{3} = \frac{2\pi}{3}}$. In the third quadrant, the angle $x$ is given by ${\pi + \frac{\pi}{3} = \frac{4\pi}{3}}$. Therefore, the solutions to the equation ${\cos x = -\frac{1}{2}}$ in the interval $[0, 2\pi)$ are ${\frac{2\pi}{3}}$ and ${\frac{4\pi}{3}}$. These are the angles that, when plugged into the original equation, will satisfy the equality. Finding these solutions requires a good understanding of the unit circle and the behavior of trigonometric functions in different quadrants. It’s a combination of geometric intuition and algebraic manipulation that leads to the final answer. By systematically considering the quadrants and reference angles, we can accurately identify all solutions within the specified interval.

Expressing the Solutions in Radians

The solutions to the equation $4 \sin^2 x = 5 + 4 \cos x$ within the interval $[0, 2\pi)$ are ${\frac{2\pi}{3}}$ and ${\frac{4\pi}{3}}$. These solutions are expressed in radians, which is the standard unit of angular measure in mathematics, particularly in trigonometry and calculus. Radians provide a natural way to relate angles to the circumference of a circle. One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. A full circle, which is 360 degrees, corresponds to $2\pi$ radians. Expressing angles in radians is crucial for several reasons. Firstly, it simplifies many mathematical formulas, especially in calculus where derivatives and integrals of trigonometric functions are more easily expressed using radians. Secondly, radians provide a more direct relationship between angles and lengths, making them more intuitive in many contexts. In the context of this problem, the solutions ${\frac{2\pi}{3}}$ and ${\frac{4\pi}{3}}$ represent angles that are fractions of the full circle. ${\frac{2\pi}{3}}$ is two-thirds of ${\pi}$ radians, which is 120 degrees, and ${\frac{4\pi}{3}}$ is four-thirds of ${\pi}$ radians, which is 240 degrees. These angles are significant points on the unit circle and are commonly encountered in trigonometric problems. Expressing solutions in radians allows for a more precise and mathematically consistent representation of angles.

Final Answer

In conclusion, by employing trigonometric identities, algebraic manipulation, and a solid understanding of the unit circle, we have successfully solved the equation $4 \sin^2 x = 5 + 4 \cos x$ within the interval $[0, 2\pi)$. The solutions, expressed in radians, are ${\frac{2\pi}{3}}$ and ${\frac{4\pi}{3}}$. These solutions represent the angles that satisfy the given equation within the specified interval. The process involved transforming the equation into a quadratic form, solving for ${\cos x}$, and then identifying the angles that correspond to the solution. This problem exemplifies the interplay between algebra and trigonometry, showcasing how algebraic techniques can be used to solve trigonometric equations. The ability to solve such equations is crucial in various fields, including physics, engineering, and computer science, where trigonometric functions are used to model periodic phenomena. The solutions ${\frac{2\pi}{3}}$ and ${\frac{4\pi}{3}}$ are specific instances of a broader class of solutions that would exist if we did not restrict the interval to $[0, 2\pi)$. However, within this interval, these are the unique angles that satisfy the equation. This exercise highlights the importance of specifying intervals when solving trigonometric equations, as the periodic nature of trigonometric functions leads to infinitely many solutions otherwise. Thus, the final answer, ${x = \frac{2\pi}{3}, \frac{4\pi}{3}}$, encapsulates the complete solution set for the given equation within the specified domain.