Problem 71 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\cos 2 x=\cos x$$

Step-by-Step Solution

Answer

The solutions for the equation within the interval \([0,2 \pi)\) are \(x = \pi/3\), \(x = \pi\), and \(x = 5\pi/3\).

Step 1: Substitution of the identity

We substitute the identity \( \cos 2x = 2 \cos^2 x - 1\) into our equation. This gives us \(2 \cos^2 x - 1 = \cos x\).

Step 2: Rewrite the equation in quadratic form

Next, rewrite the equation into a quadratic form by setting the equation to 0: \(2 \cos^2 x - \cos x - 1 = 0 \). This is a quadratic equation in \(\cos x\).

Step 3: Solve the quadratic equation

Solve the quadratic equation. You may either factor directly if it's obvious or use the quadratic formula: \(\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). By substituting \(a = 2\), \(b = -1\), and \(c = -1\) into the formula gives: \(\cos x = \frac{1 \pm \sqrt{1 + 8}}{4}\). This simplifies to: \(\cos x = 1/2\) or \(\cos x = -1\).

Step 4: Solve for \(x\)

Lastly, solve for \(x\) within the given interval. For \(\cos x = 1/2\), the solutions are \(x = \pi/3\) or \(x = 5\pi/3\). For \(\cos x = -1\), the solution is \(x = \pi\). Hence the solutions for the equation on the interval \([0, 2 \pi)\) are \(x = \pi/3\), \(x = \pi\), and \(x = 5\pi/3\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry. Often abbreviated as "cos," it's vital in understanding the relationships between the angles and sides of triangles. The cosine of an angle in a right triangle specifically refers to the ratio of the adjacent side to the hypotenuse.Understanding the cosine function is crucial when dealing with trigonometric identities and equations. For this exercise, we use the double angle identity for cosine, which states: \[ \cos 2x = 2 \cos^2 x - 1 \]This identity helps simplify expressions involving cosine and allows us to manipulate the equation into a form that can be solved. Key uses of cosine include:

• Solving right triangle problems.
• Analyzing oscillations, such as waves.
• Converting complex expressions in trigonometry into simpler equations.

In this context, understanding the properties of the cosine function and its identities enables you to solve the equation more effectively.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, generically expressed as:\[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. In this problem, we transform our trigonometric equation into a quadratic equation with regard to \(\cos x\).Our transformed equation looks like:\[2 \cos^2 x - \cos x - 1 = 0\]Here, \(a = 2\), \(b = -1\), and \(c = -1\). This quadratic form allows us to apply either factoring methods or the quadratic formula.The quadratic formula is especially useful:\[\cos x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps us find the roots of the equation, which for our scenario gives the values where \(\cos x = 1/2\) or \(\cos x = -1\). Once you have these roots, you can solve for \(x\) in the context of trigonometric functions.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation for given trigonometric values. This procedure is essential in many fields of science and engineering whenever rotational and periodic patterns arise.For this exercise, after transforming the trigonometric equation to a quadratic form and obtaining its solution, we now solve:

• For \(\cos x = 1/2\), the solutions within the interval \([0, 2\pi)\) are \(x = \pi/3\) and \(x = 5\pi/3\).
• For \(\cos x = -1\), the solution is \(x = \pi\).

When solving these equations, it's vital to remember the unit circle and the periodic nature of cosine. Cosine values repeat themselves every \(2\pi\) radians, so it’s crucial to consider all possible values for \(x\) within the given interval.Techniques like recognizing standard angle values and using trigonometric identities simplifies solving equations significantly. Practicing these methods ensures accuracy and boosts confidence when approaching similar problems. Always ensure the solutions fall within the specified interval to obtain the correct answers. This step-by-step method enhances your problem-solving skills in trigonometry.