We are given the equation:

\[
2\cos^2x - \cos x = 1
\]

Let's solve it step by step.

### Step 1: Move all terms to one side of the equation
We'll start by moving the 1 from the right side to the left side:

\[
2\cos^2x - \cos x - 1 = 0
\]

### Step 2: Factor the quadratic equation
We can factor this equation as a quadratic in terms of \(\cos x\). Let \(y = \cos x\), so the equation becomes:

\[
2y^2 - y - 1 = 0
\]

Now, factor the quadratic equation:

\[
(2y + 1)(y - 1) = 0
\]

### Step 3: Solve for \(y\)
Set each factor equal to zero:

1. \(2y + 1 = 0\)
   
   \[
   y = -\frac{1}{2}
   \]

2. \(y - 1 = 0\)

   \[
   y = 1
   \]

### Step 4: Solve for \(x\)
Recall that \(y = \cos x\), so now we solve:

1. \(\cos x = -\frac{1}{2}\)

   \[
   x = \cos^{-1}\left(-\frac{1}{2}\right)
   \]

   The solutions within the interval \(0 \leq x < 2\pi\) are:

   \[
   x = \frac{2\pi}{3}, \frac{4\pi}{3}
   \]

2. \(\cos x = 1\)

   \[
   x = \cos^{-1}(1)
   \]

   The solution within the interval \(0 \leq x < 2\pi\) is:

   \[
   x = 0
   \]

### Final Answer:
The solutions are:

\[
x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}
\]

Would you like more details on any step or have any questions?

---

Here are 5 follow-up questions related to the problem:

1. How can we derive the general solutions for \(\cos x = -\frac{1}{2}\) beyond the interval \(0 \leq x < 2\pi\)?
2. What are the implications of the periodic nature of the cosine function on the solutions?
3. How can we solve similar trigonometric equations involving sine or tangent functions?
4. What are the differences in solving equations involving \(\cos^2x\) versus \(\sin^2x\)?
5. How does the factorization process work for quadratic equations in trigonometric functions?

**Tip:** Always check the interval or domain restrictions when solving trigonometric equations to ensure you find all relevant solutions.

Learn how to solve the trigonometric equation 2cos^2x - cosx = 1 step-by-step, using quadratic factorization and cosine inverse function. Explore trigonometry concepts such as solving for x using cos x, factoring, and identifying solutions within the interval 0 ≤ x < 2π. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation