Solve: 6cos^2x+13cosx+5=0

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Introduction

Trigonometric equations are a fundamental part of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will focus on solving the trigonometric equation 6cos^2x+13cosx+5=0. This equation is a quadratic equation in terms of cosx, and we will use various techniques to solve it.

Understanding the Equation

The given equation is 6cos^2x+13cosx+5=0. This is a quadratic equation in terms of cosx, where the coefficients are 6, 13, and 5. To solve this equation, we can use various techniques such as factoring, completing the square, or using the quadratic formula.

Factoring the Equation

One way to solve the equation is to factor it. We can start by looking for two numbers whose product is 6*5=30 and whose sum is 13. These numbers are 10 and 3, so we can write the equation as:

6cos^2x+13cosx+5 = (2cosx+5)(3cosx+1) = 0

Solving for cosx

Now that we have factored the equation, we can set each factor equal to zero and solve for cosx. We have two possible solutions:

2cosx+5 = 0 --> cosx = -5/2

3cosx+1 = 0 --> cosx = -1/3

Checking the Solutions

We need to check if these solutions are valid. We can do this by plugging them back into the original equation. If the equation is true, then the solution is valid.

For cosx = -5/2, we have:

6(-5/2)^2+13(-5/2)+5 = 6(25/4)-65/2+5 = 75/2-65/2+5 = 10/2+5 = 5+5 = 10

This is not equal to zero, so cosx = -5/2 is not a valid solution.

For cosx = -1/3, we have:

6(-1/3)^2+13(-1/3)+5 = 6(1/9)-13/3+5 = 2/3-13/3+5 = -11/3+5 = -11/3+15/3 = 4/3

This is not equal to zero, so cosx = -1/3 is not a valid solution.

Using the Quadratic Formula

Another way to solve the equation is to use the quadratic formula. The quadratic formula is:

x = (-b ± √(b^2-4ac)) / 2a

In this case, a = 6, b = 13, and c = 5. Plugging these values into the formula, we get:

cosx = (-13 ± √(13^2-465)) / 2*6 cosx = (-13 ± √(169-120)) / 12 cosx = (-13 ± √49) / 12 cosx = (-13 ± 7) / 12

Solving for cosx

We have two possible solutions:

cosx = (-13 + 7) / 12 = -6/12 = -1/2

cosx = (-13 - 7) / 12 = -20/12 = -5/3

Checking the Solutions

We need to check if these solutions are valid. We can do this by plugging them back into the original equation. If the equation is true, then the solution is valid.

For cosx = -1/2, we have:

6(-1/2)^2+13(-1/2)+5 = 6(1/4)-13/2+5 = 3/2-13/2+5 = -10/2+5 = -5+5 = 0

This is equal to zero, so cosx = -1/2 is a valid solution.

For cosx = -5/3, we have:

6(-5/3)^2+13(-5/3)+5 = 6(25/9)-65/3+5 = 150/9-65/3+5 = 50/3-65/3+5 = -15/3+5 = -5+5 = 0

This is equal to zero, so cosx = -5/3 is a valid solution.

Conclusion

In this article, we have solved the trigonometric equation 6cos^2x+13cosx+5=0 using various techniques such as factoring and the quadratic formula. We have found two valid solutions: cosx = -1/2 and cosx = -5/3. These solutions satisfy the original equation and are therefore valid.

Final Answer

Introduction

In our previous article, we solved the trigonometric equation 6cos^2x+13cosx+5=0 using various techniques such as factoring and the quadratic formula. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve 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 and can be solved using various techniques.

Q: What are the common techniques used to solve trigonometric equations?

A: The common techniques used to solve trigonometric equations include:

  • Factoring: This involves expressing the equation as a product of two or more factors.
  • Completing the square: This involves rewriting the equation in a form that allows us to easily solve for the variable.
  • Using the quadratic formula: This involves using the quadratic formula to solve for the variable.
  • Using trigonometric identities: This involves using trigonometric identities to simplify the equation and solve for the variable.

Q: How do I know which technique to use?

A: The technique you use will depend on the specific equation you are trying to solve. You may need to try out different techniques to see which one works best.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x = (-b ± √(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 identify the values of a, b, and c in the equation. Then, you can plug these values into the quadratic formula and solve for the variable.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin^2x + cos^2x = 1
  • tanx = sinx / cosx
  • cotx = cosx / sinx

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

A: To use trigonometric identities to solve a trigonometric equation, you need to identify the trigonometric functions in the equation and use the identities to simplify the equation and 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 checking the solutions to see if they are valid
  • Not using the correct technique to solve the equation
  • Not simplifying the equation before solving for the variable

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug the solution back into the original equation and see if it is true. If the equation is true, then the solution is valid.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve trigonometric equations. We have covered topics such as factoring, completing the square, using the quadratic formula, and using trigonometric identities. We have also provided some common mistakes to avoid and tips for checking if a solution is valid.

Final Answer

The final answer is that solving trigonometric equations requires a deep understanding of trigonometric functions and identities, as well as the ability to use various techniques such as factoring, completing the square, and using the quadratic formula.