Problem 19 - Precalculus with Limits

Question

In Exercises 11-24, solve the equation. \( 4 \cos^2 x - 1 = 0 \)

Step-by-Step Solution

Answer

The solutions are \(x = \cos^{-1}(0.5)\), \(x = - \cos^{-1}(0.5)\) + \(2\pi\), \(x = \cos^{-1}(-0.5)\), and \(x = - \cos^{-1}(-0.5)\) + \(2\pi\)

Step 1: Isolate the trigonometric term

First, isolate the cosine term by adding 1 to both sides of the equation: \(4\cos^2 x - 1 + 1 = 0 + 1\)This simplifies to:\(4\cos^2 x = 1\)

Step 2: Solve for the cosine of x

Next, divide both sides by 4 to solve for \(\cos^2 x\):\(4\cos^2 x / 4 = 1 / 4\)This simplifies to:\(\cos^2 x = 0.25\)

Step 3: Apply the square root property

Next, take the square root of each side to solve for cos(x). Remember to include both the positive and negative square roots because we're dealing with a squared term:\(\cos x = \pm \sqrt{0.25}\)This simplifies to:\(\cos x = \pm 0.5\)

Step 4: Find the possible angles

Now that you have the possible cosine values, you can find the corresponding angle using the cosine inverse function. Remember that cosine has a positive value in the first and fourth quadrants, and negative in the second and third quadrants.So the angle solutions are:\(x = \cos^{-1}(0.5)\), \(x = - \cos^{-1}(0.5)\) + \(2\pi\) for the positive root. And for the negative root: \(x = \cos^{-1}(-0.5)\), \(x = - \cos^{-1}(-0.5)\) + \(2\pi\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. In more abstract terms, this function is used in various branches of mathematics and physics to represent periodic oscillations.

When analyzing the equation \( 4 \cos^2 x - 1 = 0 \), the term \( \cos^2 x \) represents the square of the cosine function. The cosine function, defined as \( \cos(theta) \), oscillates between -1 and 1 for any angle \(theta\). This is key when solving trigonometric equations because it sets the bounds for possible solutions.

Moreover, in the context of the unit circle, the cosine of an angle corresponds to the x-coordinate of a point on the circumference of the circle. This geometrical representation helps students to visualize the solutions on a unit circle, where the angle \(x\) is measured in radians or degrees, and the cosine represents the horizontal displacement from the origin.

Square Root Property

The square root property is an essential algebraic principle used to solve equations where a variable is squared. It states that if \( A^2 = B \), then \( A \) can be either \(\sqrt{B}\) or \( -\sqrt{B}\), since squaring both positive and negative values yields a positive result.

When applying this property to the equation \( \cos^2 x = 0.25 \), we take the square root of both sides to find the potential values of \( \cos x \). Consequently, \( \cos x \) equals both \( 0.5 \) and \( -0.5 \) as \( (\pm 0.5)^2 = 0.25 \). It is crucial not to forget the negative root since trigonometric equations often have multiple solutions within a given interval.

Understanding the square root property allows students to deal with quadratic forms of trigonometric equations effectively, leading to a comprehensive set of potential solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \cos^{-1} \), are used to find the angle associated with a known cosine value. These functions are the inverses of the trigonometric functions and provide angles as outputs. The inverse cosine function, also called arccosine, yields results ranging from 0 to \( \pi \) radians when dealing with positive inputs, and from \( \pi \) to \( 2\pi \) radians for negative inputs.

After determining that \( \cos x = \pm 0.5 \), we proceed to find the corresponding angles: \( x = \cos^{-1}(0.5) \) and \( x = \cos^{-1}(-0.5) \). But since the cosine function is periodic with a period of \( 2\pi \), it's important to consider all angles that can produce the cosines \( \pm 0.5 \) within the solution's domain, which in this case is the set of all real numbers.

By incorporating the inverse cosine function into their toolkit, students can confidently convert known ratios back into angle measures, an essential step in solving trigonometric equations. While calculators typically provide principal values, understanding the full range of angles assertively expands the solution set within the given parameters, contextualizing the conceptual framework of periodicity in trigonometry.