Solve The Trigonometric Equation For All Values $0 \leq X \ \textless \ 2\pi$.$3 \tan X = \sqrt{3}$

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the trigonometric equation $3 \tan x = \sqrt{3}$ for all values $0 \leq x \ \textless \ 2\pi$. We will break down the solution into manageable steps, using various trigonometric identities and properties to simplify the equation.

Understanding the Equation

The given equation is $3 \tan x = \sqrt{3}$. To solve this equation, we need to isolate the variable $x$. The first step is to divide both sides of the equation by 3, which gives us $\tan x = \frac{\sqrt{3}}{3}$.

Using Trigonometric Identities

We can use the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$ to rewrite the equation as $\frac{\sin x}{\cos x} = \frac{\sqrt{3}}{3}$. This identity allows us to express the tangent function in terms of sine and cosine.

Simplifying the Equation

To simplify the equation, we can multiply both sides by $\cos x$, which gives us $\sin x = \frac{\sqrt{3}}{3} \cos x$. This equation is still not in a form that is easy to solve, so we need to use another trigonometric identity.

Using the Pythagorean Identity

The Pythagorean identity states that $\sin^2 x + \cos^2 x = 1$. We can use this identity to rewrite the equation as $\left(\frac{\sqrt{3}}{3} \cos x\right)^2 + \cos^2 x = 1$. Expanding the left-hand side of the equation, we get $\frac{1}{3} \cos^2 x + \cos^2 x = 1$.

Solving for Cosine

Combining like terms, we get $\frac{4}{3} \cos^2 x = 1$. To solve for $\cos x$, we can divide both sides of the equation by $\frac{4}{3}$, which gives us $\cos^2 x = \frac{3}{4}$. Taking the square root of both sides, we get $\cos x = \pm \frac{\sqrt{3}}{2}$.

Finding the Values of X

Now that we have found the values of $\cos x$, we can use the inverse cosine function to find the values of $x$. The inverse cosine function is denoted by $\cos^{-1} x$, and it returns the angle whose cosine is equal to the given value.

Using the Inverse Cosine Function

Using the inverse cosine function, we can find the values of $x$ as follows:

• $\cos x = \frac{\sqrt{3}}{2}$: $x = \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$
• $\cos x = -\frac{\sqrt{3}}{2}$: $x = \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) = \frac{11\pi}{6}$

Checking the Solutions

To check the solutions, we can substitute the values of $x$ back into the original equation and verify that they satisfy the equation.

Conclusion

In this article, we have solved the trigonometric equation $3 \tan x = \sqrt{3}$ for all values $0 \leq x \ \textless \ 2\pi$. We have used various trigonometric identities and properties to simplify the equation and find the values of $x$. The solutions are $x = \frac{\pi}{6}$ and $x = \frac{11\pi}{6}$.

Final Answer

The final answer is $\boxed{\frac{\pi}{6}, \frac{11\pi}{6}}$.

Additional Resources

• Trigonometric Equations
• Inverse Cosine Function
• Trigonometric Identities

Step-by-Step Solution

1. Divide both sides of the equation by 3: $\tan x = \frac{\sqrt{3}}{3}$
2. Use the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$ to rewrite the equation: $\frac{\sin x}{\cos x} = \frac{\sqrt{3}}{3}$
3. Multiply both sides by $\cos x$: $\sin x = \frac{\sqrt{3}}{3} \cos x$
4. Use the Pythagorean identity to rewrite the equation: $\left(\frac{\sqrt{3}}{3} \cos x\right)^2 + \cos^2 x = 1$
5. Expand the left-hand side of the equation: $\frac{1}{3} \cos^2 x + \cos^2 x = 1$
6. Combine like terms: $\frac{4}{3} \cos^2 x = 1$
7. Divide both sides by $\frac{4}{3}$: $\cos^2 x = \frac{3}{4}$
8. Take the square root of both sides: $\cos x = \pm \frac{\sqrt{3}}{2}$
9. Use the inverse cosine function to find the values of $x$: $x = \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$ and $x = \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) = \frac{11\pi}{6}$
   Solving Trigonometric Equations: A Comprehensive Guide ===========================================================

Q&A: Solving Trigonometric Equations

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects or the behavior of electrical circuits.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the variable, which is usually represented by x. You can use various trigonometric identities and properties to simplify the equation and find the values of x.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 x + \cos^2 x = 1$
• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$

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

A: The Pythagorean identity states that $\sin^2 x + \cos^2 x = 1$. You can use this identity to rewrite the equation and simplify it. For example, if you have the equation $\sin^2 x + \cos^2 x = 4$, you can use the Pythagorean identity to rewrite it as $1 = 4$, which is a contradiction.

Q: What is the inverse cosine function?

A: The inverse cosine function is denoted by $\cos^{-1} x$, and it returns the angle whose cosine is equal to the given value. For example, if you have the equation $\cos x = \frac{\sqrt{3}}{2}$, you can use the inverse cosine function to find the value of x.

Q: How do I use the inverse cosine function to solve a trigonometric equation?

A: To use the inverse cosine function to solve a trigonometric equation, you need to isolate the variable, which is usually represented by x. You can then use the inverse cosine function to find the value of x.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

• Not isolating the variable
• Not using the correct trigonometric identities
• Not checking the solutions

Q: How do I check the solutions to a trigonometric equation?

A: To check the solutions to a trigonometric equation, you need to substitute the values of x back into the original equation and verify that they satisfy the equation.

Q: What are some real-world applications of trigonometric equations?

A: Some real-world applications of trigonometric equations include:

• Modeling the motion of objects
• Analyzing the behavior of electrical circuits
• Solving problems in physics and engineering

Additional Resources

• Trigonometric Equations
• Inverse Cosine Function
• Trigonometric Identities

Step-by-Step Solution

1. Divide both sides of the equation by 3: $\tan x = \frac{\sqrt{3}}{3}$
2. Use the trigonometric identity $\tan x = \frac{\sin x}{\cos x}$ to rewrite the equation: $\frac{\sin x}{\cos x} = \frac{\sqrt{3}}{3}$
3. Multiply both sides by $\cos x$: $\sin x = \frac{\sqrt{3}}{3} \cos x$
4. Use the Pythagorean identity to rewrite the equation: $\left(\frac{\sqrt{3}}{3} \cos x\right)^2 + \cos^2 x = 1$
5. Expand the left-hand side of the equation: $\frac{1}{3} \cos^2 x + \cos^2 x = 1$
6. Combine like terms: $\frac{4}{3} \cos^2 x = 1$
7. Divide both sides by $\frac{4}{3}$: $\cos^2 x = \frac{3}{4}$
8. Take the square root of both sides: $\cos x = \pm \frac{\sqrt{3}}{2}$
9. Use the inverse cosine function to find the values of x: $x = \cos^{-1} \left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6}$ and $x = \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) = \frac{11\pi}{6}$

Final Answer

The final answer is $\boxed{\frac{\pi}{6}, \frac{11\pi}{6}}$.