Solve The Equation For X X X : 2 Cot ⁡ X = 1 + Tan ⁡ X Where 0 ≤ X ≤ 360 ∘ 2 \cot X = 1 + \tan X \quad \text{where} \quad 0 \leq X \leq 360^{\circ} 2 Cot X = 1 + Tan X Where 0 ≤ X ≤ 36 0 ∘

$$

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $2 \cot x = 1 + \tan x$ where $0 \leq x \leq 360^{\circ}$. We will break down the solution into manageable steps, using various trigonometric identities and properties to simplify the equation and find the value of $x$.

Understanding the Equation

The given equation is $2 \cot x = 1 + \tan x$. To solve this equation, we need to first understand the properties of the trigonometric functions involved. The cotangent function is defined as $\cot x = \frac{\cos x}{\sin x}$, and the tangent function is defined as $\tan x = \frac{\sin x}{\cos x}$. We can rewrite the equation using these definitions:

$$2 \frac{\cos x}{\sin x} = 1 + \frac{\sin x}{\cos x}$$

Simplifying the Equation

To simplify the equation, we can start by multiplying both sides by $\sin x \cos x$ to eliminate the fractions:

$$2 \cos^2 x = \sin^2 x + \sin x \cos x$$

We can further simplify the equation by using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Rearranging the terms, we get:

$$2 \cos^2 x - \sin^2 x = \sin x \cos x$$

Using Trigonometric Identities

We can use the double-angle formula for sine to simplify the equation:

$$\sin 2x = 2 \sin x \cos x$$

Substituting this into the equation, we get:

$$2 \cos^2 x - \sin^2 x = \sin 2x$$

Solving for $x$

To solve for $x$, we can use the fact that $\sin 2x = 2 \sin x \cos x$. We can rewrite the equation as:

$$2 \cos^2 x - \sin^2 x = 2 \sin x \cos x$$

Dividing both sides by $2 \cos x$, we get:

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

We can further simplify the equation by using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. Rearranging the terms, we get:

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

Simplifying the equation, we get:

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

Combining like terms, we get:

$$\cos x - \frac{1}{2 \cos x} + \frac{\cos x}{2} = \sin x$$

Simplifying the equation, we get:

$$\frac{3 \cos x}{2} - \frac{1}{2 \cos x} = \sin x$$

Finding the Value of $x$

To find the value of $x$, we can use the fact that $\sin x = \frac{1}{2}$. Substituting this into the equation, we get:

$$\frac{3 \cos x}{2} - \frac{1}{2 \cos x} = \frac{1}{2}$$

Multiplying both sides by $2 \cos x$, we get:

$$3 \cos^2 x - 1 = \cos x$$

Rearranging the terms, we get:

$$3 \cos^2 x - \cos x - 1 = 0$$

We can solve this quadratic equation using the quadratic formula:

$$\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substituting the values of $a$, $b$, and $c$, we get:

$$\cos x = \frac{1 \pm \sqrt{(-1)^2 - 4(3)(-1)}}{2(3)}$$

Simplifying the equation, we get:

$$\cos x = \frac{1 \pm \sqrt{13}}{6}$$

Conclusion

In this article, we solved the equation $2 \cot x = 1 + \tan x$ where $0 \leq x \leq 360^{\circ}$. We used various trigonometric identities and properties to simplify the equation and find the value of $x$. The final solution is $\cos x = \frac{1 \pm \sqrt{13}}{6}$. We hope this article has provided a clear and concise guide to solving trigonometric equations.

Final Answer

$$

Q: What is the first step in solving a trigonometric equation?

A: The first step in solving a trigonometric equation is to simplify the equation using various trigonometric identities and properties. This may involve rewriting the equation in terms of sine and cosine, or using the Pythagorean identity to eliminate the square root.

Q: How do I use the Pythagorean identity to simplify a trigonometric equation?

A: The Pythagorean identity states that $\sin^2 x + \cos^2 x = 1$. You can use this identity to eliminate the square root by rearranging the terms. For example, if you have the equation $\sin^2 x - \cos^2 x = 0$, you can rewrite it as $(\sin^2 x + \cos^2 x) - 2 \cos^2 x = 0$, which simplifies to $1 - 2 \cos^2 x = 0$.

Q: What is the double-angle formula for sine?

A: The double-angle formula for sine states that $\sin 2x = 2 \sin x \cos x$. You can use this formula to simplify trigonometric equations by substituting $\sin 2x$ for $2 \sin x \cos x$.

Q: How do I solve a quadratic equation in terms of cosine?

A: To solve a quadratic equation in terms of cosine, you can use the quadratic formula:

$$\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Substitute the values of $a$, $b$, and $c$ into the formula, and simplify to find the value of $\cos x$.

Q: What is the final answer to the equation $2 \cot x = 1 + \tan x$?

A: The final answer to the equation $2 \cot x = 1 + \tan x$ is $\cos x = \frac{1 \pm \sqrt{13}}{6}$.

Q: How do I check my solution to a trigonometric equation?

A: To check your solution to a trigonometric equation, you can substitute the value of $x$ back into the original equation and simplify. If the equation is true, then your solution is correct.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

• The Pythagorean identity: $\sin^2 x + \cos^2 x = 1$
• The double-angle formula for sine: $\sin 2x = 2 \sin x \cos x$
• The double-angle formula for cosine: $\cos 2x = 2 \cos^2 x - 1$
• The sum and difference formulas for sine and cosine: $\sin (x + y) = \sin x \cos y + \cos x \sin y$ and $\sin (x - y) = \sin x \cos y - \cos x \sin y$

Q: How do I use trigonometric identities to simplify a trigonometric equation?

A: To use trigonometric identities to simplify a trigonometric equation, you can substitute the identity into the equation and simplify. For example, if you have the equation $\sin^2 x - \cos^2 x = 0$, you can rewrite it as $(\sin^2 x + \cos^2 x) - 2 \cos^2 x = 0$, which simplifies to $1 - 2 \cos^2 x = 0$.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

• Simplify the equation using trigonometric identities and properties
• Use the Pythagorean identity to eliminate the square root
• Use the double-angle formula for sine and cosine to simplify the equation
• Check your solution by substituting the value of $x$ back into the original equation
• Use trigonometric identities to simplify the equation and find the value of $x$