Problem 124 - Algebra and Trigonometry

Question

solve each equation on the interval \([0,2 \pi) .\) $$ 3 \cos ^{2} x-\sin x=\cos ^{2} x $$

Step-by-Step Solution

Answer

First simplify the equation to form a quadratic equation in terms of cosine. Then, convert the resulting equation into its standard form by making an appropriate substitution. Solve the quadratic equation to find the values of cosine that satisfy the equation. Lastly, utilize the arccosine function to find the solutions within the given range.

Step 1: Simplify the Equation

The given equation is: \(3 \cos ^{2} x-\sin x=\cos ^{2} x\). Let's group the cosine terms together: \(3 \cos ^{2} x - \cos ^{2} x = \sin x\). This simplifies to \(2 \cos ^{2} x = \sin x\). To simplify further, convert \(\sin x\) to \(1 - \cos ^{2} x\). Therefore, the resulting equation is: \(2 \cos ^{2} x = 1 - \cos ^{2} x\). Rearrange this into a quadratic equation: \(3 \cos ^{2} x + \sin x - 1 = 0.\)

Step 2: Convert to Standard Quadratic Form

The equation \(3 \cos ^{2} x + \sin x - 1 = 0\) is similar to the quadratic equation in the form \(ax^{2} + bx + c = 0\). Let \(y = \cos x\). Substitute this into the equation, which is now \(3y^{2} + y - 1 = 0\). This structure is closer to a standard format.

Step 3: Solve Quadratic Equation

Now solve the quadratic equation \(3y^{2} + y - 1 = 0\). This can be done by factoring, quadratic formula, or by completing the square. Solving this equation will give us the values for \(y\) which is equal to \(\cos x\).

Step 4: Find the Roots of the Equation

Once the values for \(y = \cos x\) are calculated, use an inverse cosine function \(\arccos\) to find the corresponding \(x\) values. Since \(\cos x = y\), then \(x = \arccos y\). Remember that the solutions have to be in the range from \(0\) to \(2 \pi\).

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos x\), is one of the primary trigonometric functions. It is often used to describe the change in horizontal position over a cycle in a unit circle. The function outputs values between -1 and 1, and these values are repeated in cycles. In problems like the one given, the cosine function is used to find relationships between angles and their cosine values. Understanding how \(\cos x\) behaves helps in solving trigonometric equations effectively, especially when manipulating the equation to isolate cosine-related terms.

For example, recognizing the identity \(\sin^2 x = 1 - \cos^2 x\) is crucial for simplifying trigonometric expressions. This ability to transform different trigonometric terms into similar forms makes solving equations much more manageable.

Quadratic Equations

Quadratic equations come in the form \(ax^2 + bx + c = 0\). Solving such equations can be done using several methods: factoring, using the quadratic formula, or completing the square. The aim is to find the values of \(x\) that satisfy the equation.

The problem turns into a quadratic form when expressed as \(3 \cos^2 x + \sin x - 1 = 0\). By letting \(y = \cos x\), the equation becomes \(3y^2 + y - 1 = 0\). This highlights how trigonometric equations can often be solved using algebraic techniques by recognizing underlying patterns and similarities with quadratic equations.

Inverse Trigonometric Functions

Inverse trigonometric functions help to find angles when given a trigonometric ratio. In the case of the cosine function, the inverse is \(\arccos y\). This function provides an angle whose cosine value is \(y\).

After finding \(y = \cos x\) in the quadratic solution, using \(x = \arccos y\) determines the specific angles within the defined interval \([0, 2\pi)\). Remember that cosine is positive in both the first and fourth quadrants, so the inverse cosine will give solutions within these ranges.

Trigonometric Identities

Trigonometric identities are crucial in simplifying and working with trigonometric expressions. They express one trigonometric function in terms of others. For instance, the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) allows us to express \(\sin x\) as \(1 - \cos^2 x\).

This identity played an essential role in the problem by transforming \(\sin x\) into terms of \(\cos x\), leading to a more straightforward quadratic equation. Mastery of these identities enables easier manipulation of trigonometric equations and aids in finding solutions efficiently.