Given: $3 \tan 4x = -2 \cos 4x$1. Without Using A Calculator, Show That $\sin 4x = -0.5$ Is The Only Solution To The Above Equation.2. Hence, Determine The General Solution Of $x$ In The Equation $3 \tan 4x = -2 \cos

Introduction

Trigonometric equations are an essential part of mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $3 \tan 4x = -2 \cos 4x$ without using a calculator. We will first show that $\sin 4x = -0.5$ is the only solution to the equation and then determine the general solution of $x$.

Step 1: Simplify the Equation

The given equation is $3 \tan 4x = -2 \cos 4x$. To simplify the equation, we can use the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

3 \tan 4x = -2 \cos 4x
3 \frac{\sin 4x}{\cos 4x} = -2 \cos 4x
3 \sin 4x = -2 \cos^2 4x

Step 2: Use the Pythagorean Identity

We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to rewrite the equation.

3 \sin 4x = -2 \cos^2 4x
3 \sin 4x = -2 (1 - \sin^2 4x)
3 \sin 4x = -2 + 2 \sin^2 4x
2 \sin^2 4x - 3 \sin 4x - 2 = 0

Step 3: Solve the Quadratic Equation

The equation $2 \sin^2 4x - 3 \sin 4x - 2 = 0$ is a quadratic equation in terms of $\sin 4x$. We can solve it using the quadratic formula.

\sin 4x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\sin 4x = \frac{3 \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}
\sin 4x = \frac{3 \pm \sqrt{9 + 16}}{4}
\sin 4x = \frac{3 \pm \sqrt{25}}{4}
\sin 4x = \frac{3 \pm 5}{4}

Step 4: Find the Solutions

We have two possible solutions for $\sin 4x$:

\sin 4x = \frac{3 + 5}{4} = \frac{8}{4} = 2
\sin 4x = \frac{3 - 5}{4} = \frac{-2}{4} = -0.5

However, we know that the range of the sine function is $[-1, 1]$, so the solution $\sin 4x = 2$ is not valid.

Conclusion

Therefore, we have shown that $\sin 4x = -0.5$ is the only solution to the equation $3 \tan 4x = -2 \cos 4x$.

General Solution

To find the general solution of $x$, we can use the inverse sine function:

4x = \sin^{-1} (-0.5)
4x = -\frac{\pi}{6}
x = -\frac{\pi}{24}

However, we know that the sine function has a period of $2\pi$, so we can add multiples of $2\pi$ to the solution to get all possible solutions:

x = -\frac{\pi}{24} + \frac{2\pi k}{4}
x = -\frac{\pi}{24} + \frac{\pi k}{2}

where $k$ is an integer.

Final Answer

Therefore, the general solution of $x$ is:

x = -\frac{\pi}{24} + \frac{\pi k}{2}

Q&A: Frequently Asked Questions

Q: What is the main goal of solving trigonometric equations? A: The main goal of solving trigonometric equations is to find the values of the variables that satisfy the equation.

Q: What are some common trigonometric equations? A: Some common trigonometric equations include:

• $\sin \theta = \frac{1}{2}$
• $\cos \theta = \frac{\sqrt{3}}{2}$
• $\tan \theta = 1$

Q: How do I simplify trigonometric equations? A: To simplify trigonometric equations, you can use the following steps:

1. Use the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
2. Use the sum and difference formulas: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
3. Use the double-angle formulas: $\sin 2\theta = 2\sin \theta \cos \theta$
4. Use the half-angle formulas: $\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

Q: How do I solve quadratic equations in trigonometric form? A: To solve quadratic equations in trigonometric form, you can use the quadratic formula:

$\sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: What is the general solution of a trigonometric equation? A: The general solution of a trigonometric equation is the set of all possible solutions, including the principal solution and all other solutions that are obtained by adding multiples of the period to the principal solution.

Q: How do I find the principal solution of a trigonometric equation? A: To find the principal solution of a trigonometric equation, you can use the inverse trigonometric function:

$\theta = \sin^{-1} (\sin \theta)$

Q: What is the period of a trigonometric function? A: The period of a trigonometric function is the length of the interval over which the function repeats itself.

Q: How do I find the period of a trigonometric function? A: To find the period of a trigonometric function, you can use the following formulas:

• For the sine and cosine functions: $T = \frac{2\pi}{k}$
• For the tangent function: $T = \pi$

where $k$ is the coefficient of the variable in the function.

Q: What is the range of a trigonometric function? A: The range of a trigonometric function is the set of all possible output values of the function.

Q: How do I find the range of a trigonometric function? A: To find the range of a trigonometric function, you can use the following formulas:

• For the sine and cosine functions: $[-1, 1]$
• For the tangent function: $(-\infty, \infty)$

Conclusion

Solving trigonometric equations requires a deep understanding of trigonometric functions and their properties. By using the Pythagorean identity, sum and difference formulas, double-angle formulas, and half-angle formulas, you can simplify trigonometric equations and solve them using the quadratic formula. The general solution of a trigonometric equation is the set of all possible solutions, including the principal solution and all other solutions that are obtained by adding multiples of the period to the principal solution.