Find All Solutions To The Equation Without Using A Calculator:$\[ 4 \cos^2x + 4 \cos X + 1 = 0 \\]Use The Smallest Nonnegative Angles In Your Solution. Select The Correct Answer Below, And If Necessary, Fill In The Answer Box(es) To Complete
Introduction
In this article, we will delve into the world of trigonometric equations and explore a specific equation that involves the cosine function. The equation in question is 4 cos^2x + 4 cos x + 1 = 0, and we will aim to find all solutions to this equation without relying on a calculator. We will use the smallest nonnegative angles in our solution, and we will provide a step-by-step guide to help readers understand the process.
Understanding the Equation
The given equation is a quadratic equation in terms of cos x. It can be rewritten as (2 cos x + 1)^2 = 0. This equation involves the square of a binomial, which can be factored to reveal the solutions.
Factoring the Equation
To factor the equation (2 cos x + 1)^2 = 0, we can take the square root of both sides. This gives us 2 cos x + 1 = 0. We can then isolate cos x by subtracting 1 from both sides and dividing by 2.
Solving for cos x
Solving for cos x, we get cos x = -1/2. This is the value of cos x that satisfies the equation.
Finding the Angles
To find the angles that correspond to cos x = -1/2, we can use the inverse cosine function. The inverse cosine function returns an angle in radians between 0 and pi. We can use this function to find the angles that satisfy the equation.
Using the Inverse Cosine Function
Using the inverse cosine function, we get x = arccos(-1/2). This gives us the principal angle that satisfies the equation.
Finding the Smallest Nonnegative Angles
To find the smallest nonnegative angles, we can use the periodicity of the cosine function. The cosine function has a period of 2pi, which means that the values of cos x repeat every 2pi. We can use this property to find the smallest nonnegative angles that satisfy the equation.
Finding the Angles in Quadrants I and IV
To find the angles in quadrants I and IV, we can add multiples of 2pi to the principal angle. This gives us the angles in quadrants I and IV that satisfy the equation.
Finding the Angles in Quadrants II and III
To find the angles in quadrants II and III, we can subtract multiples of 2pi from the principal angle. This gives us the angles in quadrants II and III that satisfy the equation.
Conclusion
In conclusion, we have found the solutions to the equation 4 cos^2x + 4 cos x + 1 = 0. We have used the smallest nonnegative angles in our solution, and we have provided a step-by-step guide to help readers understand the process. The solutions to the equation are x = 2pi/3, x = 4pi/3, x = 2pi + 2pi/3, and x = 2pi + 4pi/3.
Final Answer
The final answer is:
Introduction
In our previous article, we explored the trigonometric equation 4 cos^2x + 4 cos x + 1 = 0 and found the solutions to this equation. In this article, we will provide a Q&A section to help readers understand the process and address any questions they may have.
Q: What is the equation 4 cos^2x + 4 cos x + 1 = 0?
A: The equation 4 cos^2x + 4 cos x + 1 = 0 is a quadratic equation in terms of cos x. It can be rewritten as (2 cos x + 1)^2 = 0.
Q: How do I solve the equation 4 cos^2x + 4 cos x + 1 = 0?
A: To solve the equation 4 cos^2x + 4 cos x + 1 = 0, you can start by factoring the equation. This gives you (2 cos x + 1)^2 = 0. You can then take the square root of both sides to get 2 cos x + 1 = 0. Finally, you can isolate cos x by subtracting 1 from both sides and dividing by 2.
Q: What is the value of cos x that satisfies the equation 4 cos^2x + 4 cos x + 1 = 0?
A: The value of cos x that satisfies the equation 4 cos^2x + 4 cos x + 1 = 0 is -1/2.
Q: How do I find the angles that correspond to cos x = -1/2?
A: To find the angles that correspond to cos x = -1/2, you can use the inverse cosine function. The inverse cosine function returns an angle in radians between 0 and pi. You can use this function to find the principal angle that satisfies the equation.
Q: What are the smallest nonnegative angles that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0?
A: The smallest nonnegative angles that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0 are x = 2pi/3 and x = 4pi/3.
Q: How do I find the angles in quadrants I and IV that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0?
A: To find the angles in quadrants I and IV that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0, you can add multiples of 2pi to the principal angle. This gives you the angles in quadrants I and IV that satisfy the equation.
Q: How do I find the angles in quadrants II and III that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0?
A: To find the angles in quadrants II and III that satisfy the equation 4 cos^2x + 4 cos x + 1 = 0, you can subtract multiples of 2pi from the principal angle. This gives you the angles in quadrants II and III that satisfy the equation.
Conclusion
In conclusion, we have provided a Q&A section to help readers understand the process of solving the trigonometric equation 4 cos^2x + 4 cos x + 1 = 0. We hope that this section has been helpful in addressing any questions you may have had.
Final Answer
The final answer is: