Problem 10 - Calculus with Concepts in Calculus

Question

Solve the equation for \(x\) in \([0,2 \pi)\). $$ \cos 2 x=\cos x $$

Step-by-Step Solution

Answer

Solutions are \(x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}\).

Step 1: Use Trigonometric Identity for Cosine Double Angle

Firstly, we will make use of the trigonometric identity for cosine of a double angle: \[\cos 2x = 2\cos^2 x - 1\]Substitute this into the equation:\[2\cos^2 x - 1 = \cos x\]

Step 2: Rearrange the Equation

Next, rearrange the equation into a standard quadratic form:\[2\cos^2 x - \cos x - 1 = 0\]This resembles a quadratic equation in terms of \(\cos x\).

Step 3: Solve the Quadratic Equation

Let \(y = \cos x\). The equation becomes:\[2y^2 - y - 1 = 0\]Solve for \(y\) using the quadratic formula:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 2\), \(b = -1\), \(c = -1\). Substitute these values:\[y = \frac{1 \pm \sqrt{(-1)^2 - 4 \times 2 \times (-1)}}{4}\]\[y = \frac{1 \pm \sqrt{1 + 8}}{4}\]\[y = \frac{1 \pm \sqrt{9}}{4}\]\[y = \frac{1 \pm 3}{4}\]The solutions for \(y\) are \(y = 1\) and \(y = -\frac{1}{2}\).

Step 4: Find the Solutions for x

If \(\cos x = 1\), then \(x = 0\).If \(\cos x = -\frac{1}{2}\), the values of \(x\) within \([0, 2\pi)\) are:\[x = \frac{2\pi}{3}\] and \[x = \frac{4\pi}{3}\].

Step 5: Verify the Solutions are in the Given Domain

Finally, ensure all solutions are in the interval \([0, 2\pi)\). The solutions from the previous steps, \(x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}\), all lie within this interval.

Key Concepts

Cosine Double Angle Identity

The Cosine Double Angle Identity is a useful trigonometric identity used to simplify expressions involving the cosine of a double angle. It states that \(\cos 2x = 2\cos^2 x - 1\).
This identity helps in transforming an equation with a double angle into a more manageable form.
By substituting \(\cos 2x\) with \(2\cos^2 x - 1\), we can create a simpler or more solvable equation.
For instance, in the problem \(\cos 2x = \cos x\), substituting \(2\cos^2 x - 1\) for \(\cos 2x\) gives us \(2\cos^2 x - 1 = \cos x\).

• This transformation allows us to leverage algebraic methods such as factoring or using the quadratic formula.
• It's key in solving trigonometric equations, particularly those involving angles of the form \(2x\).

Understanding and applying this identity is fundamental in tackling many trigonometric problems efficiently.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, often written in the standard form \(ax^2 + bx + c = 0\). In trigonometric contexts, we often see its derivatives like \(2\cos^2 x - \cos x - 1 = 0\), which resembles a quadratic form.
To solve this type of equation, identifying it as a quadratic allows us to apply specific techniques such as:

• The quadratic formula, \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which provides solutions for \(y\).
• Factoring, when applicable, to break down the quadratic into simpler linear factors.

In our case, we substitute \(y = \cos x\) to get \(2y^2 - y - 1 = 0\).

Using the quadratic formula with \(a = 2\), \(b = -1\), \(c = -1\), helps find solutions for \(y\), which are \( y = 1\) and \( y = -\frac{1}{2}\).

Recognizing the quadratic structure in trigonometric problems is pivotal for applying systematic methods to find the solution.

Solution Interval

The Solution Interval refers to the domain within which the solutions for \(x\) must lie, particularly in trigonometric problems. Such specificity is crucial because trigonometric functions are periodic.
In the exercise given, the interval is \([0, 2\pi)\), which represents one complete cycle of the cosine function in radians.
Once we solve the equation for \(\cos x\), we obtain potential \(x\)-values and must verify them against the interval.

• This means checking each solution to confirm that it falls within the specified range.
• For \(\cos x = 1\), the solution \(x = 0\) is included in \([0, 2\pi)\).
• For \(\cos x = -\frac{1}{2}\), solutions \(x = \frac{2\pi}{3}\) and \(x = \frac{4\pi}{3}\) also fit within the interval.

Inspecting solution intervals ensures that we adhere to the problem's constraints, which is essential for reaching the correct answer in periodic problems.