Solve The Equation: ${ 3 \sin^2 X - 5 \sin X = 2 }$

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 a specific type of trigonometric equation, namely the quadratic equation involving sine. We will use the equation ${ 3 \sin^2 x - 5 \sin x = 2 }$ as a case study and provide a step-by-step guide on how to solve it.

Understanding the Equation

Before we dive into the solution, let's understand the equation and its components. The equation is a quadratic equation in terms of $\sin x$, where the coefficient of $\sin^2 x$ is 3, the coefficient of $\sin x$ is -5, and the constant term is 2. Our goal is to find the values of $x$ that satisfy this equation.

Step 1: Rearrange the Equation

To solve the equation, we need to rearrange it to a standard quadratic form. We can do this by adding 2 to both sides of the equation, which gives us:

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

This is a quadratic equation in terms of $\sin x$, and we can use the quadratic formula to solve it.

Step 2: Apply the Quadratic Formula

The quadratic formula is given by:

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

In our case, $a = 3$, $b = -5$, and $c = 2$. Plugging these values into the formula, we get:

$$\sin x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(2)}}{2(3)}$$

Simplifying the expression, we get:

$$\sin x = \frac{5 \pm \sqrt{25 - 24}}{6}$$

$$\sin x = \frac{5 \pm \sqrt{1}}{6}$$

$$\sin x = \frac{5 \pm 1}{6}$$

This gives us two possible values for $\sin x$, namely $\frac{5 + 1}{6} = \frac{6}{6} = 1$ and $\frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}$.

Step 3: Find the Values of $x$

Now that we have the values of $\sin x$, we can find the corresponding values of $x$. We know that $\sin x = 1$ when $x = \frac{\pi}{2}$, and $\sin x = \frac{2}{3}$ when $x = \sin^{-1} \left(\frac{2}{3}\right)$.

Conclusion

In this article, we solved the trigonometric equation ${ 3 \sin^2 x - 5 \sin x = 2 }$ using the quadratic formula. We rearranged the equation to a standard quadratic form, applied the quadratic formula, and found the values of $x$ that satisfy the equation. This problem illustrates the importance of understanding the components of a trigonometric equation and using the appropriate techniques to solve it.

Common Mistakes to Avoid

When solving trigonometric equations, it's essential to avoid common mistakes such as:

• Not rearranging the equation to a standard quadratic form
• Not applying the quadratic formula correctly
• Not considering the domain of the trigonometric function

Tips and Tricks

Here are some tips and tricks to help you solve trigonometric equations:

• Always rearrange the equation to a standard quadratic form
• Use the quadratic formula to solve the equation
• Consider the domain of the trigonometric function
• Use a calculator or software to check your solutions

Real-World Applications

Trigonometric equations have numerous real-world applications, including:

• Physics: Trigonometric equations are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
• Computer Science: Trigonometric equations are used in computer graphics, game development, and other areas of computer science.

Conclusion

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 describe the relationships between the angles and side lengths of triangles.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the trigonometric function and then use the inverse trigonometric function to find the value of the angle. You can also use the quadratic formula to solve quadratic trigonometric equations.

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 apply the quadratic formula to a trigonometric equation?

A: To apply the quadratic formula to a trigonometric equation, you need to rearrange the equation to a standard quadratic form, identify the coefficients of the quadratic equation, and then plug them into the quadratic formula.

Q: What are some common trigonometric equations?

A: Some common trigonometric equations include:

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

Q: How do I find the values of $x$ that satisfy a trigonometric equation?

A: To find the values of $x$ that satisfy a trigonometric equation, you need to use the inverse trigonometric function to find the angle that corresponds to the given value of the trigonometric function.

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

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

• Physics: Trigonometric equations are used to describe the motion of objects in terms of position, velocity, and acceleration.
• Engineering: Trigonometric equations are used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
• Computer Science: Trigonometric equations are used in computer graphics, game development, and other areas of computer science.

Q: How do I avoid common mistakes when solving trigonometric equations?

A: To avoid common mistakes when solving trigonometric equations, you need to:

• Always rearrange the equation to a standard quadratic form
• Use the quadratic formula to solve the equation
• Consider the domain of the trigonometric function
• Use a calculator or software to check your solutions

Q: What are some tips and tricks for solving trigonometric equations?

A: Some tips and tricks for solving trigonometric equations include:

• Always use the quadratic formula to solve quadratic trigonometric equations
• Consider the domain of the trigonometric function
• Use a calculator or software to check your solutions
• Practice, practice, practice!

Conclusion

In conclusion, solving trigonometric equations is a crucial skill for students and professionals alike. By understanding the components of a trigonometric equation and using the appropriate techniques, we can solve even the most complex equations. Remember to avoid common mistakes, use the quadratic formula, and consider the domain of the trigonometric function. With practice and patience, you'll become proficient in solving trigonometric equations and apply them to real-world problems.