Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$4 \cos ^{2} x-1=0$$

Step-by-Step Solution

Answer

Solutions in radians: \(\frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}\). Solutions in degrees: \(60^\circ, 120^\circ, 240^\circ, 300^\circ\).

Step 1: Identify and Transform the Equation

The given equation is \(4 \cos^2 x - 1 = 0\). Notice that this can be rearranged as a difference of squares: \( (2 \cos x - 1)(2 \cos x + 1) = 0 \). Thus, we can set up two equations: \(2 \cos x - 1 = 0\) and \(2 \cos x + 1 = 0\).

Step 2: Solve for Cosine Values

First, solve \(2 \cos x - 1 = 0\) which simplifies to \(\cos x = \frac{1}{2}\). Next, solve \(2 \cos x + 1 = 0\) which simplifies to \(\cos x = -\frac{1}{2}\). These are the two possible cosine values that satisfy the original equation.

Step 3: Find Solutions in Radians

For \(\cos x = \frac{1}{2}\), the angles are \(x = \frac{\pi}{3}, \ \frac{5\pi}{3}\) within the interval \([0, 2\pi)\). For \(\cos x = -\frac{1}{2}\), the angles are \(x = \frac{2\pi}{3}, \ \frac{4\pi}{3}\).

Step 4: Convert Solutions to Degrees

Convert the radian solutions to degrees: \(\frac{\pi}{3} = 60^\circ\), \(\frac{5\pi}{3} = 300^\circ\), \(\frac{2\pi}{3} = 120^\circ\), and \(\frac{4\pi}{3} = 240^\circ\). Thus, all solutions in degrees are: \(60^\circ, 120^\circ, 240^\circ, 300^\circ\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often abbreviated as "cos". It relates the angle in a right triangle to the ratio of the adjacent side over the hypotenuse. The range of the cosine function is from -1 to 1.
To recall, the cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Therefore, its value cycles between -1 and 1 as the angle varies from 0 to 2\(\pi\).

In the exercise, the equation \( 4 \cos^2 x - 1 = 0 \) uses the square of the cosine function, where transformations reveal simpler equations, \( \cos x = \frac{1}{2} \) and \( \cos x = -\frac{1}{2} \), which are recognizable values based on standard angles in trigonometry.

Angle Measures

Angles can be measured in different units, mainly degrees and radians. Each type of measurement has its place in both theoretical and applied mathematics.

• Degrees: This is a measure of angle based on a circle divided into 360 equal parts
• Radians: Defined as the angle subtended by an arc of a circle that has the same length as the circle's radius

When solving trigonometric equations like our exercise, determining specific angle measures is critical. The standard practice is to find the angles that correspond to the given trigonometric value. Here, \( \cos x = \frac{1}{2} \) corresponds to angles like \(60^\circ\) or \( \frac{\pi}{3} \), while \( \cos x = -\frac{1}{2} \) corresponds to \(120^\circ\) or \(\frac{2\pi}{3} \), keeping the solutions within the least nonnegative limits.

Radian and Degree Conversion

Conversion between radians and degrees is an essential skill in trigonometry, allowing navigation between different contexts.
The formula for converting from radians to degrees and vice versa is derived from the relation \( 180^\circ = \pi \text{ radians} \). This leads to the conversion formulas:

• Degrees to Radians: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
• Radians to Degrees: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)

Using these conversions in our problem, we take the angles found in radians, like \(\frac{\pi}{3}\), and convert them to degrees, yielding results like \(60^\circ\). This step is crucial, especially when instructions demand the answers in the least nonnegative angle measures.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for any angle. They are vital tools in transforming and solving trigonometric problems.
One common identity is the quadratic form \( a \cos^2 x + b \cos x + c = 0 \), which can be rearranged and solved like the simple quadratic equation seen in algebra.

In our exercise, the expression \( 4 \cos^2 x - 1 = 0 \) is an example of using identities to simplify and find solutions. It is based on transforming the equation using the basic identity: \(\cos(2\theta) = 2\cos^2(\theta) - 1\), reworked in our problem.Applying these identities helps find and verify valid solutions, ensuring they align with known trigonometric values on the unit circle.