Problem 34 - Algebra 2

Question

Solve each equation for all values of \(\theta\) if \(\theta\) is measured in radians. \(4 \cos ^{2} \theta-4 \cos \theta+1=0\)

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\), \(k \in \mathbb{Z}\).

Step 1: Identify the Quadratic Form

The given equation, \(4 \cos^2 \theta - 4 \cos \theta + 1 = 0\), is in the form of a quadratic equation in terms of \(\cos \theta\). Treat \(\cos \theta\) as a variable, for example, let \(x = \cos \theta\). This transforms the equation to \(4x^2 - 4x + 1 = 0\).

Step 2: Use the Quadratic Formula

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 4\), \(b = -4\), and \(c = 1\), we proceed to solve for \(x\). Substitute these values into the formula: \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 4 \times 1}}{2 \times 4}\).

Step 3: Simplify the Solution

Simplify the quadratic formula: \(x = \frac{4 \pm \sqrt{16 - 16}}{8}\) which further simplifies to \(x = \frac{4 \pm 0}{8}\). Thus, \(x = \frac{4}{8} = \frac{1}{2}\). Therefore, \(\cos \theta = \frac{1}{2}\).

Step 4: Solve for \(\theta\) in Given Range

The equation \(\cos \theta = \frac{1}{2}\) has solutions \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) within the range of \([0, 2\pi]\). These are standard angles where the cosine is \(\frac{1}{2}\).

Step 5: General Solution for \(\theta\)

For a complete set of solutions, include all angles where \(\cos \theta = \frac{1}{2}\). This occurs at \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\), where \(k\) is any integer.

Key Concepts

Quadratic Form

A quadratic form refers to a polynomial expression of degree two. Typically, it is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. In the given exercise, the equation is in terms of \(\cos \theta\), meaning it's a quadratic equation where \(x = \cos \theta\).
When you encounter such an equation, it's useful to temporarily substitute \(\cos \theta\) with another variable, like \(x\). This substitution transforms the equation into a recognizable quadratic form, making it easier to handle. Once the equation is solved, you revert back to \(\cos \theta\) to find the actual trigonometric solutions.

Quadratic Formula

The quadratic formula is a powerful tool in solving quadratic equations. It’s given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps in finding the values of \(x\) that satisfy the quadratic equation. Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
In our context, after substituting \(\cos \theta\) with \(x\), the equation \(4x^2 - 4x + 1 = 0\) comes to life. By applying the quadratic formula, we calculate:

• \(a = 4\)
• \(b = -4\)
• \(c = 1\)

Substituting these values into the formula yields the roots of the equation. These roots, \(x = \frac{1}{2}\), represent the potential values of \(\cos \theta\), meaning the angle \(\theta\) must be determined, where the cosine takes this value.

Cosine Function

The cosine function is a fundamental trigonometric function. It is defined for an angle \(\theta\) in a right-angled triangle, as the ratio of the adjacent side to the hypotenuse.
When considering the unit circle, \(\cos \theta\) represents the x-coordinate of the point where the terminal side of an angle \(\theta\), in standard position, intersects the circle's circumference. For the problem in question, solving \(\cos \theta = \frac{1}{2}\) involves identifying angles that provide this cosine value. Commonly, these angles, found through the unit circle, are \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\) within the principal range \([0, 2\pi]\).
These angles are well-known in trigonometry for their cosine values and can be verified by understanding their corresponding positions on the unit circle.

Solutions in Radians

Radians are a way to measure angles, based on the radius of a circle. One complete revolution around a circle is \(2\pi\) radians, which is equivalent to 360 degrees.
In trigonometry, using radians is preferred over degrees because they simplify the mathematics involved in calculus and analysis. When solving equations like the one given, results are typically expressed in radians.If we want the general solution for \(\cos \theta = \frac{1}{2}\), periodicity must be considered. The cosine function repeats every \(2\pi\) radians, so solutions expand beyond the basic interval of \([0, 2\pi]\) using:

• \(\theta = \frac{\pi}{3} + 2k\pi\)
• \(\theta = \frac{5\pi}{3} + 2k\pi\)

Here \(k\) is an integer, ensuring we include all possible angles that reflect the given cosine value. This gives a comprehensive understanding of solving trigonometric equations in radians.