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 a combination of trigonometric identities and algebraic manipulations.

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 the sine and cosine functions.

Simplifying the Equation

We can simplify the equation by multiplying both sides by $\cos x$, which gives us $\sin x = -\frac{\sqrt{3}}{3} \cos x$. This equation is still in terms of the sine and cosine functions, but it is now in a more manageable form.

Using the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to rewrite the equation as $\sin^2 x + \cos^2 x = \left(-\frac{\sqrt{3}}{3} \cos x\right)^2$. This identity allows us to express the sum of the squares of the sine and cosine functions in terms of a single variable.

Solving for x

We can simplify the equation by expanding the right-hand side, which gives us $\sin^2 x + \cos^2 x = \frac{1}{3} \cos^2 x$. We can then subtract $\cos^2 x$ from both sides, which gives us $\sin^2 x = -\frac{2}{3} \cos^2 x$.

Using the Quadratic Formula

We can use the quadratic formula to solve for $\cos x$. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, we have $a = -\frac{2}{3}$, $b = 0$, and $c = 1$. Plugging these values into the quadratic formula, we get $\cos x = \frac{0 \pm \sqrt{0 - 4\left(-\frac{2}{3}\right)\left(1\right)}}{2\left(-\frac{2}{3}\right)}$.

Simplifying the Quadratic Formula

We can simplify the quadratic formula by evaluating the expression under the square root. This gives us $\cos x = \frac{0 \pm \sqrt{\frac{8}{3}}}{-\frac{4}{3}}$.

Solving for x

We can simplify the expression by multiplying both sides by $-\frac{3}{4}$, which gives us $\cos x = \mp \frac{\sqrt{2}}{2}$. We can then use the inverse cosine function to solve for $x$. The inverse cosine function is given by $x = \cos^{-1} \left(\mp \frac{\sqrt{2}}{2}\right)$.

Finding the Solutions

We can use a calculator or a trigonometric table to find the values of $x$ that satisfy the equation. The solutions are given by $x = \cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$ and $x = \pi + \cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$.

Conclusion

In this article, we solved the trigonometric equation $3 \tan x = -\sqrt{3}$ for all values $0 \leq x \ \textless \ 2\pi$. We used a combination of trigonometric identities and algebraic manipulations to isolate the variable $x$. The solutions are given by $x = \cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$ and $x = \pi + \cos^{-1} \left(-\frac{\sqrt{2}}{2}\right)$. We hope that this article has provided a clear and concise explanation of how to solve trigonometric equations.

Additional Resources

For more information on trigonometric equations, we recommend the following resources:

• Trigonometric Equations
• Solving Trigonometric Equations
• Trigonometric Identities

Final Thoughts

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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 a combination of trigonometric identities and algebraic manipulations to solve the equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

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

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

A: The Pythagorean identity is $\sin^2 x + \cos^2 x = 1$. You can use this identity to rewrite the equation in terms of a single variable, such as $\sin x$ or $\cos x$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula to solve a trigonometric equation?

A: To use the quadratic formula to solve a trigonometric equation, you need to rewrite the equation in the form of a quadratic equation. You can then use the quadratic formula to solve for the variable.

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

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

• Not using the correct trigonometric identity
• Not isolating the variable
• Not checking the solutions
• Not using the correct method to solve the equation

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

A: To check your solutions to a trigonometric equation, you need to plug the solutions back into the original equation. If the solutions satisfy the equation, then they are valid solutions.

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
• Modeling population growth and decay

Q: How do I practice solving trigonometric equations?

A: To practice solving trigonometric equations, you can try the following:

• Work on practice problems
• Use online resources, such as Khan Academy or MIT OpenCourseWare
• Join a study group or find a study partner
• Take online courses or attend workshops

Conclusion

Solving trigonometric equations is a crucial skill for students and professionals alike. In this article, we provided a step-by-step guide on how to solve trigonometric equations, as well as some frequently asked questions and answers. We hope that this article has provided a clear and concise explanation of how to solve trigonometric equations.