Problem 47 - Algebra and Trigonometry

Question

Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$ 4 \cos ^{2} x-1=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(4 \cos ^{2} x-1=0\) within the interval \([0,2 \pi)\) are \(x = \frac{\pi}{3}\), \(x = \frac{2 \pi}{3}\), \(x = \frac{4 \pi}{3}\), and \(x = \frac{5 \pi}{3}\).

Step 1: Rewrite the equation in standard form

The given equation is \(4 \cos ^{2} x-1=0\). To make it easier to work with, rewrite it in standard form: \(4 \cos ^{2} x = 1\) Divide both sides by 4: \(\cos ^{2} x = \frac{1}{4}\)

Step 2: Apply the basic trigonometric identity

To solve for \(x\), take the square root of both sides. Remember, when you take square root of a square, there should be two results: a positive root and a negative root. These leads to: \(\cos x = \pm \frac{1}{2}\) This means either \(\cos x = \frac{1}{2}\) or \(\cos x = -\frac{1}{2}\)

Step 3: Solve for \(x\) in the given interval

To solve for \(x\), consider the unit circle where each point has coordinates \((cos x, sin x)\). \(\cos x = \frac{1}{2}\) at \(x = \frac{\pi}{3}\) and \(x = \frac{5 \pi}{3}\), while \(\cos x = -\frac{1}{2}\) at \(x = \frac{2 \pi}{3}\) and \(x = \frac{4 \pi}{3}\). Therefore, \(x\) can be any of these four values within the interval \([0,2 \pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities play a crucial role in simplifying and solving trigonometric equations. These are equations that involve trigonometric functions and are true for all values of the variables involved. When tackling the equation 4 \( cos^2 x - 1\) = 0, we use a fundamental identity known as the Pythagorean identity, which relates the square of the sine function to the square of the cosine function: \(sin^2 x + cos^2 x = 1\).

For the given problem, after isolating the cosine term and taking its square root, we exploit the fact that \(cos^2 x = \(cos x\)^2\), meaning the equation represents the square of cosine. This allows us to apply the basic trigonometric identity \(cos^2 x + sin^2 x = 1\) to find that \(cos x = \pm\sqrt{\frac{1}{4}}\), or \(cos x = \pm\frac{1}{2}\).

Remember that there are many trigonometric identities, including sum-difference formulas, double-angle formulas, and half-angle formulas, all of which may be required to solve more complex trigonometric equations. Understanding how to manipulate these identities is key to solving the equations efficiently and correctly.

Unit Circle

The unit circle is a fundamental concept in trigonometry and provides a geometric interpretation of trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. Any point on the unit circle can be defined by \( (cos \theta, sin \theta) \), where \( \theta \) is the angle formed by the line segment from the origin to the point and the positive x-axis.

In the context of the solved equation \(4 cos^2 x - 1 = 0\), once we determine \(cos x = \pm\frac{1}{2}\), we can refer to the unit circle to find the angles (or \(x\)) that correspond to these cosine values. Because the unit circle allows us to see all the angles that correspond to a particular sine or cosine value, we can accurately determine that the angles associated with \(cos x = \(\frac{1}{2}\)\) are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\), while the angles for \(cos x = -\frac{1}{2}\) are \(x = \frac{2\pi}{3}\) and \(x = \frac{4\pi}{3}\).

By understanding the unit circle, students can visualize and solve trigonometric equations more effectively since it provides a clear depiction of the relationship between angles and trigonometric function values.

Quadratic Equations in Trigonometry

Finding Solutions to Quadratic Trigonometric Equations

Similar to standard quadratic equations, quadratic equations in trigonometry can generally be expressed in the form \( a(trig \; function)^2 + b(trig \; function) + c = 0 \), where \(trig \; function\) represents a trigonometric function such as sine or cosine. To solve these equations, one approach is to use trigonometric identities to simplify them into a more recognizable quadratic form, as we did by isolating \(cos^2 x\) in the example problem.

Once the equation is in a solvable form, the next step is to find the values of the trigonometric function that satisfy the equation, which usually involves taking the square root of both sides. This process can yield two solutions because of the inherent plus-or-minus nature of the square root. As we saw, \(cos x = \pm\frac{1}{2}\).

Interpreting Results within a Specific Interval

It is also crucial to interpret these solutions within a given interval, often in the context of the unit circle. For example, we solve for all values of \(x\) within the interval \( [0,2\pi) \), ensuring we find all possible angles that satisfy the quadratic equation. Understanding how to find and interpret these solutions is essential for solving trigonometric equations that can be represented in a quadratic form.