Problem 27 - Precalculus Essentials

Question

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\cos 4 x=-\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

The solutions to the equation in the interval [0,2π) are \(x=\frac{5\pi}{24}, \frac{13\pi}{24}, \frac{21\pi}{24}, \frac{29\pi}{24}, \frac{7\pi}{24}, \frac{15\pi}{24}, \frac{23\pi}{24}, \frac{31\pi}{24}\).

Step 1: Compare to Known Values of Cosine

Remember that \(\cos(\theta)= -\frac{\sqrt{3}}{2}\) when \(\theta=\frac{5\pi}{6}\) or \(\theta=\frac{7\pi}{6}\). Hence, for our problem, we equate the cosine equation to \(\frac{5\pi}{6}\) and \(\frac{7\pi}{6}\).

Step 2: Isolate Variable

As the equation given is \(\cos 4x = -\frac{\sqrt{3}}{2}\) we need to isolate for \(x\). Equation: \(4x=\frac{5\pi}{6} + 2\pi n\) or \(4x=\frac{7\pi}{6} + 2\pi n\), where \(n\) is an integer. Then solve for \(x\), \(x=\frac{5\pi}{24} + \frac{\pi n}{2}\) or \(x=\frac{7\pi}{24} + \frac{\pi n}{2}\).

Step 3: List All Solutions in the Interval [0,2π)

Now we have to list all the solutions for \(x\) in the specified interval [0,2π) by substituting integer values for \(n\). For \(x=\frac{5\pi}{24} + \frac{\pi n}{2}\), the solutions are \(x=\frac{5\pi}{24}, \frac{13\pi}{24}, \frac{21\pi}{24}, \frac{29\pi}{24}\). For \(x=\frac{7\pi}{24} + \frac{\pi n}{2}\), the solutions are \(x=\frac{7\pi}{24}, \frac{15\pi}{24}, \frac{23\pi}{24}, \frac{31\pi}{24}\).

Key Concepts

Multiple Angle Identities

When dealing with trigonometric equations like \( \cos 4x = -\frac{\sqrt{3}}{2} \), multiple angle identities come into play. This involves recognizing angles that are multiplied by a factor, such as 4 in this case. Multiple angle identities are essential because they allow us to transform complicated trigonometric expressions into more manageable ones.

For example, knowing that \( \cos(\theta) = -\frac{\sqrt{3}}{2} \) at specific angles helps solve equations with multiples of those angles by equating them to known values like \( \frac{5\pi}{6} \) or \( \frac{7\pi}{6} \). This lets you rewrite the original problem as two simpler equations for \(x\). Thus, understanding multiple angle identities is key to simplifying and solving these kinds of equations.

Cosine Function

The cosine function, one of the fundamental trigonometric functions, relates the angle in a right triangle to the length of the adjacent side over the hypotenuse. When dealing with trigonometric equations, the cosine function helps identify the angle for given values.

In our exercise, we use the fact that the cosine of certain angles equals \(-\frac{\sqrt{3}}{2}\). This knowledge comes from understanding cosine's behavior on the unit circle. Cosine values repeat periodically, making it possible to find an infinite number of solutions for trigonometric equations. Exploring how cosine changes over a full rotation (\(0\) to \(2\pi\)) helps us find all possible angles that satisfy the equation.

Interval Notation

Interval notation is a concise way to represent a range of values, often used with angles in trigonometry. It's especially useful when specifying where solutions need to fall, like in the interval \([0, 2\pi)\).

In this context, the brackets \([\) and \()\) indicate whether the endpoints are included or not. The closed bracket \([\) includes \(0\), whereas the open bracket \()\) excludes \(2\pi\). This ensures that solutions wrap around the circle without repeating the starting angle, providing a complete set of solutions within one period.

Solutions to Trigonometric Equations

Finding solutions to trigonometric equations often involves comparing to known angle values, isolating variables, and understanding periodicity.
The exercise requires solving the equation \( \cos 4x = -\frac{\sqrt{3}}{2} \) by using known solutions for \(\cos \theta\) and adjusting them for the multiple angle of \(4x\).

• First, equate \(4x\) to angles \(\frac{5\pi}{6}\) and \(\frac{7\pi}{6}\).
• Next, solve for \(x\) by dividing the equation by 4 and considering the periodic nature by adding multiples of \(2\pi\).
• Finally, identify all values of \(x\) that fit within the specified interval, ensuring no solutions are repeated.

Understanding how these solutions manifest in cyclic patterns aids in catching every possible answer within an interval.