Solving Trigonometric Equations Tan X + 3 Cot X = 5 Sec X And Cos 2x = Cos X

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In the realm of mathematics, trigonometric equations hold a significant place, often appearing in various fields such as physics, engineering, and computer graphics. These equations involve trigonometric functions like sine, cosine, tangent, and their reciprocals. Solving these equations requires a strong understanding of trigonometric identities, algebraic manipulation, and problem-solving techniques. This article aims to provide a comprehensive guide on tackling two specific trigonometric equations: ${ \tan x + 3 \cot x = 5 \sec x }$ and ${ \cos 2x = \cos x }$. We will delve into the step-by-step solutions, underlying concepts, and potential pitfalls to avoid, ensuring a clear and thorough understanding for readers of all levels.

1. Solving ${ \tan x + 3 \cot x = 5 \sec x }$

1.1 Initial Transformation: Expressing in Terms of Sine and Cosine

The first step in solving any trigonometric equation is often to express all functions in terms of sine and cosine. This simplifies the equation and allows us to use fundamental trigonometric identities. Let's start by rewriting the given equation ${ \tan x + 3 \cot x = 5 \sec x }$ using the definitions of tangent, cotangent, and secant:

• ${ \tan x = \frac{\sin x}{\cos x} }$
• ${ \cot x = \frac{\cos x}{\sin x} }$
• ${ \sec x = \frac{1}{\cos x} }$

Substituting these into the original equation, we get:

${ \frac{\sin x}{\cos x} + 3 \frac{\cos x}{\sin x} = 5 \frac{1}{\cos x} }$

1.2 Eliminating Fractions: Multiplying by ${ \sin x \cos x }$

To eliminate the fractions, we multiply both sides of the equation by ${ \sin x \cos x }$. This step is crucial in simplifying the equation into a more manageable form. Note that multiplying by ${ \sin x \cos x }$ implies that ${ \sin x \neq 0 }$ and ${ \cos x \neq 0 }$, as these values would make the denominators zero in the original equation. We will need to check for these as extraneous solutions later.

Multiplying both sides by ${ \sin x \cos x }$, we obtain:

${ \sin^2 x + 3 \cos^2 x = 5 \sin x }$

1.3 Using the Pythagorean Identity: Transforming ${ \cos^2 x }$ to ${ 1 - \sin^2 x }$

To further simplify the equation, we can use the Pythagorean identity, which states that ${ \sin^2 x + \cos^2 x = 1 }$. We can rearrange this to express ${ \cos^2 x }$ in terms of ${ \sin^2 x }$:

${ \cos^2 x = 1 - \sin^2 x }$

Substituting this into our equation, we get:

${ \sin^2 x + 3(1 - \sin^2 x) = 5 \sin x }$

Expanding and simplifying, we have:

${ \sin^2 x + 3 - 3 \sin^2 x = 5 \sin x }$

${ -2 \sin^2 x - 5 \sin x + 3 = 0 }$

1.4 Quadratic Equation: Solving for ${ \sin x }$

Now we have a quadratic equation in terms of ${ \sin x }$. To make it easier to solve, let ${ y = \sin x }$. Our equation becomes:

${ -2y^2 - 5y + 3 = 0 }$

Multiplying by -1 to simplify, we get:

${ 2y^2 + 5y - 3 = 0 }$

We can factor this quadratic equation:

${ (2y - 1)(y + 3) = 0 }$

This gives us two possible solutions for ${ y }$:

${ 2y - 1 = 0 \Rightarrow y = \frac{1}{2} }$

${ y + 3 = 0 \Rightarrow y = -3 }$

Since ${ y = \sin x }$, we have:

${ \sin x = \frac{1}{2} \quad \text{or} \quad \sin x = -3 }$

However, since the sine function's range is ${ [-1, 1] }$, ${ \sin x = -3 }$ is not a valid solution. Therefore, we only consider ${ \sin x = \frac{1}{2} }$.

1.5 Finding Solutions for ${ x }$: Using the Unit Circle

The equation ${ \sin x = \frac{1}{2} }$ has solutions in the first and second quadrants. The reference angle is ${ \frac{\pi}{6} }$, so the solutions in the interval ${ [0, 2\pi) }$ are:

${ x = \frac{\pi}{6} \quad \text{and} \quad x = \pi - \frac{\pi}{6} = \frac{5\pi}{6} }$

Therefore, the general solutions are:

${ x = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad x = \frac{5\pi}{6} + 2n\pi }$

where ${ n }$ is an integer.

1.6 Checking for Extraneous Solutions

Recall that we multiplied the original equation by ${ \sin x \cos x }$, which means we assumed ${ \sin x \neq 0 }$ and ${ \cos x \neq 0 }$. We need to check if our solutions satisfy these conditions. For ${ x = \frac{\pi}{6} }$ and ${ x = \frac{5\pi}{6} }$, neither sine nor cosine is zero, so these solutions are valid.

2. Solving ${ \cos 2x = \cos x }$

2.1 Using the Double Angle Identity: Expressing ${ \cos 2x }$ in Terms of ${ \cos x }$

The equation ${ \cos 2x = \cos x }$ involves the double angle identity for cosine. There are three forms of the double angle identity for cosine, but the most useful in this case is:

${ \cos 2x = 2 \cos^2 x - 1 }$

Substituting this into the equation, we get:

${ 2 \cos^2 x - 1 = \cos x }$

2.2 Rearranging into a Quadratic Equation: Setting the Equation to Zero

To solve for ${ \cos x }$, we rearrange the equation into a quadratic form:

${ 2 \cos^2 x - \cos x - 1 = 0 }$

2.3 Solving the Quadratic Equation: Factoring or Using the Quadratic Formula

Let ${ y = \cos x }$. The equation becomes:

${ 2y^2 - y - 1 = 0 }$

We can factor this quadratic equation:

${ (2y + 1)(y - 1) = 0 }$

This gives us two possible solutions for ${ y }$:

${ 2y + 1 = 0 \Rightarrow y = -\frac{1}{2} }$

${ y - 1 = 0 \Rightarrow y = 1 }$

Since ${ y = \cos x }$, we have:

${ \cos x = -\frac{1}{2} \quad \text{or} \quad \cos x = 1 }$

2.4 Finding Solutions for ${ x }$: Using the Unit Circle

For ${ \cos x = -\frac{1}{2} }$, the solutions in the interval ${ [0, 2\pi) }$ are in the second and third quadrants. The reference angle is ${ \frac{\pi}{3} }$, so the solutions are:

${ x = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \quad \text{and} \quad x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} }$

For ${ \cos x = 1 }$, the solution in the interval ${ [0, 2\pi) }$ is:

${ x = 0 }$

Therefore, the general solutions are:

${ x = \frac{2\pi}{3} + 2n\pi, \quad x = \frac{4\pi}{3} + 2n\pi, \quad \text{and} \quad x = 2n\pi }$

where ${ n }$ is an integer.

2.5 Verification of Solutions

In this case, we used the double angle identity and solved a quadratic equation, so there are no extraneous solutions introduced. The solutions we found are valid.

In summary, solving trigonometric equations like ${ \tan x + 3 \cot x = 5 \sec x }$ and ${ \cos 2x = \cos x }$ requires a systematic approach. This includes transforming the equations using trigonometric identities, simplifying them into manageable forms (such as quadratic equations), and finding the general solutions. It's also crucial to check for extraneous solutions, especially when multiplying or dividing by trigonometric functions. By mastering these techniques, you can confidently tackle a wide range of trigonometric equations. Trigonometric equations are a cornerstone of advanced mathematics, and proficiency in solving them opens doors to further studies in various scientific and engineering disciplines. Practice is key to mastering these concepts, and consistent effort will lead to improved skills and a deeper understanding of trigonometry. Remember, trigonometric equations may seem daunting at first, but with a clear strategy and a solid grasp of the fundamentals, they become manageable and even enjoyable to solve. Keep practicing, and you'll find yourself becoming more adept at navigating the world of trigonometry. This comprehensive guide has equipped you with the necessary tools and insights to solve these types of equations, and with continued practice, you will undoubtedly enhance your mathematical prowess. This is crucial for success in various fields, including physics, engineering, and computer science. Understanding trigonometric functions and their applications is not just an academic exercise; it is a practical skill that can be applied in numerous real-world scenarios. Whether you are designing a bridge, simulating fluid dynamics, or developing computer graphics, trigonometry plays a vital role. Therefore, investing time and effort in mastering these concepts is a worthwhile endeavor that will yield long-term benefits. Furthermore, the ability to solve trigonometric equations fosters critical thinking and problem-solving skills, which are valuable assets in any profession. The process of identifying the appropriate trigonometric identities, manipulating equations, and verifying solutions hones your analytical abilities and enhances your capacity to approach complex problems systematically. In conclusion, this detailed exploration of solving ${ \tan x + 3 \cot x = 5 \sec x }$ and ${ \cos 2x = \cos x }$ not only provides a step-by-step guide but also underscores the broader significance of trigonometry in mathematics and various practical applications. By mastering these techniques and continuing to practice, you will build a strong foundation for future mathematical endeavors and develop invaluable problem-solving skills that will serve you well in any field you pursue.