Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$3 \cos x+5=0$$

Step-by-Step Solution

Answer

There are no solutions, given that \(-\frac{5}{3}\) is outside the range of cosine.

Step 1: Isolate the Trigonometric Function

Start with the given equation: \[ 3 \cos x + 5 = 0 \]Subtract 5 from both sides of the equation to isolate the trigonometric term:\[ 3 \cos x = -5 \]

Step 2: Solve for Cosine Value

Divide each side of the equation by 3 to solve for \(\cos x\):\[ \cos x = -\frac{5}{3} \]

Step 3: Analyze the Cosine Value

Since the range of the cosine function is \([-1, 1]\), there are no real values for \(x\) that satisfy \(\cos x = -\frac{5}{3}\), as \(-\frac{5}{3}\) is outside this range.

Key Concepts

Cosine Function

The cosine function, represented as \( \cos x \), is one of the fundamental trigonometric functions. It plays a crucial role in trigonometry and is often used to solve various types of equations, from basic to complex. In geometric terms, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Mathematically, the function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians. This property makes it especially useful in solving trigonometric equations.

Key characteristics of the cosine function include:

• The maximum value is 1, which occurs at integer multiples of \(2\pi\), specifically when \(x = 0, 2\pi, 4\pi\), etc.
• The minimum value is -1, which occurs at odd multiples of \(\pi\), such as \(x = \pi, 3\pi, 5\pi\), etc.
• Cosine is an even function, meaning that \( \cos(-x) = \cos(x) \).

These properties suggest that solving equations involving cosine often involves utilizing its periodicity and symmetry. Identifying where the cosine function equals specific values is a key part of solving trigonometric equations. Understanding these concepts will help in cases where equations might have no solution in a certain range.

Range of Trigonometric Functions

When solving trigonometric equations, understanding the range of trigonometric functions is essential. The range refers to the set of possible outputs or values a function can take.

For the cosine function, the range is

• \([-1, 1]\). This means that for any real number input, the resulting cosine value will always fall between -1 and 1.

Understanding this concept helps in determining the feasibility of solutions in the context of trigonometric equations. For instance, in the problem given, we arrived at \( \cos x = -\frac{5}{3} \). Since \(-\frac{5}{3}\) is not within the range of the cosine function, it indicates that there are no real solutions over the specified interval. Any calculated cosine values falling outside of \([-1, 1]\) indicate an error or an absence of real solutions.

In summary, identifying the range of trigonometric functions as a first verification step can save time and prevent misjudgments while solving equations. This knowledge is particularly valuable in ensuring that expected outcomes are realistic and attainable.

Interval Notation

Interval notation is a powerful way of expressing the range or domain of a set of numbers, often used in mathematics to describe the solutions of equations.

In the context of trigonometric equations, interval notation helps specify the subset of real numbers we are interested in. For our problem, the interval \([0, 2\pi)\) indicates that we are looking for solutions within a full cycle of the unit circle, starting from \(0\) up to, but not including, \(2\pi\). This means:

• "\([0, 2\pi)\)" specifies that \(0\) is included (closed interval at \(0\)) but \(2\pi\) is not (open interval at \(2\pi\)).
• This type of notation is precise and critical for accurately communicating exact solution sets in trigonometry and calculus.

By using interval notation, mathematicians can succinctly convey any limitations or restrictions on the numerical domain of interest. This is instrumental in highlighting the specific range of values while avoiding unnecessary or unintended values.

With this knowledge, students can better appreciate how interval notation simplifies communication and analysis of solutions in trigonometric equations.