Problem 16 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$\cos ^{2} x+2 \cos x+1=0$$

Step-by-Step Solution

Answer

The solution is \(x = \pi\).

Step 1: Recognize the Trigonometric Equation Form

The given equation is \( \cos^2 x + 2\cos x + 1 = 0 \). Notice that this resembles a quadratic equation in the form \( a\cos^2 x + b\cos x + c = 0 \), where \( a = 1 \), \( b = 2 \), and \( c = 1 \).

Step 2: Factor the Quadratic Equation

Rewrite the equation in factored form by observing that \( \cos^2 x + 2\cos x + 1 = (\cos x + 1)^2 = 0 \). This simplification holds because squaring any expression will be zero only when the expression itself is zero.

Step 3: Solve for the Trigonometric Function

Since \((\cos x + 1)^2 = 0\) implies that \(\cos x + 1 = 0\), solve this to find \(\cos x = -1\).

Step 4: Determine the Solutions Over \([0, 2\pi)\)

Identify the angle \(x\) within the interval \([0, 2\pi)\) for which \(\cos x = -1\). The angle that satisfies this condition is \(x = \pi\).

Key Concepts

Quadratic equations

In trigonometry, it’s common to encounter equations that resemble quadratic equations. A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \). When you see a term like \( \cos^2 x \), think of it as similar to \( x^2 \) in algebra.

This structure allows us to utilize factoring techniques that we use for typical quadratic equations. Here, recognizing \( \cos^2 x + 2 \cos x + 1 = 0 \) as a form of quadratic equation is pivotal. By understanding this resemblance, we can apply the same procedures:

• Identify the coefficients \( a \), \( b \), and \( c \). In our trigonometric case, \( a = 1 \), \( b = 2 \), and \( c = 1 \).
• Factor the quadratic expression, which simplifies the equation significantly.

Factoring helps convert the equation into a simpler form, which makes it easier to solve for the trigonometric function, as it isolates the variable in question.

Cosine function

The cosine function is a key trigonometric function that appears frequently in problems involving angles and periodic functions. Cosine, denoted as \( \cos(x) \), reflects the x-coordinate of a point on the unit circle at a given angle \( x \).

In the equation \( \cos^2 x + 2 \cos x + 1 = 0 \), recognizing the properties of cosine is essential to solving the problem.

• The range of the cosine function is from -1 to 1.
• It is an even function, meaning that \( \cos(-x) = \cos(x) \).
• Cosine has specific known values at key angles, like \( \cos(0) = 1 \), \( \cos(\pi/2) = 0 \), \( \cos(\pi) = -1 \).

Using these properties, once we transform and simplify the original problem to \( \cos x = -1 \), it quickly guides us to the solution, leveraging the cosine values at standard angles.

Angle solutions

After simplifying the equation to find \( \cos x = -1 \), the next step is to determine the angle or angles that satisfy this condition within the specified interval. In trigonometry, solutions for angles often revolve around the geometry of the unit circle.

To find the solution, consider the interval \([0, 2\pi)\):

• Cosine is -1 at the angle \( x = \pi \). This is because on the unit circle, the point corresponding to \( x = \pi \) lies at (-1, 0), where cosine represents the x-coordinate.
• No other angles within this interval will give a cosine value of -1.

Identifying these specific angle solutions is crucial for understanding periodic behaviors and correctly solving trigonometric equations in the context of defined intervals.