Problem 30 - Precalculus Mathematics for Calculus

Question

Find all solutions of the given equation. $$4 \cos \theta+1=0$$

Step-by-Step Solution

Answer

\(\theta = \cos^{-1}(-1/4) + 2k\pi\) or \(\theta = 2\pi - \cos^{-1}(-1/4) + 2k\pi\), where \(k \in \mathbb{Z}\).

Step 1: Isolate the Cosine Function

The given equation is \(4\cos\theta + 1 = 0\). To isolate the cosine function, subtract 1 from both sides of the equation:\[4\cos\theta = -1\]

Step 2: Solve for Cosine

Now, divide both sides of the isolated equation by 4 in order to solve for \(\cos\theta\):\[\cos\theta = \frac{-1}{4}\]

Step 3: Determine General Solutions

Use the cosine inverse function to find the general solution of \(\theta\). Since cosine is negative in the second and third quadrants, we find:\[\theta = \cos^{-1}(-1/4) + 2k\pi\text{ or }\theta = 2\pi - \cos^{-1}(-1/4) + 2k\pi, \text{ where } k \in \mathbb{Z}\]

Step 4: Express Exact Solutions

The two general solutions can be expressed as:\[\theta = \cos^{-1}(-1/4) + 2k\pi\]\[\theta = (2\pi - \cos^{-1}(-1/4)) + 2k\pi\]where \(k \in \mathbb{Z}\). These represent all solutions of the equation taking into account the periodic nature of the cosine function.

Key Concepts

Understanding the Cosine Function

The cosine function, often represented by \( \cos \theta \), is a fundamental component of trigonometry. It relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. However, its importance stretches far beyond triangles. It is a periodic function, meaning it repeats its values in regular intervals. For cosine, these intervals are every \( 2\pi \) radians, or 360 degrees, due to its wave-like pattern.
The cosine function ranges from -1 to 1, and it mirrors symmetrically around the y-axis. One notable characteristic is its behavior in different quadrants:

• In the first quadrant \((0, \frac{\pi}{2})\), cosine is positive.
• In the second quadrant \((\frac{\pi}{2}, \pi)\), cosine becomes negative.
• It remains negative in the third quadrant \((\pi, \frac{3\pi}{2})\).
• Finally, in the fourth quadrant \((\frac{3\pi}{2}, 2\pi)\), it turns positive again.

Understanding these properties is crucial when solving trigonometric equations as they dictate where the solutions can be found depending on the nature of the cosine function in each quadrant.

Exploring Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \cos^{-1}(x) \), are essential tools for finding angles when you know the cosine value. They "undo" the basic trigonometric functions, allowing you to retrieve angles from certain ratios. In the context of the cosine function, \( \cos^{-1}(x) \) will yield an angle whose cosine is \( x \). This function is particularly vital when solving trigonometric equations.
The range of the inverse cosine, \( \cos^{-1}(x) \), is typically \([0, \pi]\). This is because, for cosine, we mainly consider the angles in the first and second quadrants where it can span all necessary values from 1 to -1. When using \( \cos^{-1} \) to solve problems, it's important to remember that this will provide solutions in terms of principal values. Additional solutions must be calculated when the full breadth of possible angles is required, often by using the properties of cosine in different quadrants as laid out in the previous section.

General Solution of Trigonometric Equations

Solving trigonometric equations involves finding all possible angles that satisfy a given equation. Since trigonometric functions are periodic, they can have infinitely many solutions. Hence, rather than just finding a single solution, we determine a general solution. This incorporates the periodicity of functions like cosine.
The general solution of an equation, such as \( \cos \theta = \frac{-1}{4} \), involves both the initial solution provided by the inverse trigonometric function and accounting for all repeats of cosine’s period \( 2\pi \). For cosine, if \( \theta = \cos^{-1}(\frac{-1}{4}) \) is one solution, another in the second quadrant follows as \( 2\pi - \cos^{-1}(\frac{-1}{4}) \).
The complete set of solutions is expressed using \( k \), an integer, to denote all possible rotations through the period:

• \( \theta = \cos^{-1} \left(-\frac{1}{4} \right) + 2k\pi \)
• \( \theta = 2\pi - \cos^{-1} \left(-\frac{1}{4} \right) + 2k\pi \)

Here, \( k \in \mathbb{Z} \) signifies every integer so that the function frames infinite solutions along the cosine wave’s cycle.