Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-54: Refer to the graph of the equation on the specified interval. Find all values of \(x\) such that for the real number \(a\), (a) \(y=a\), (b) \(y>a\), and (c) \(y

Step-by-Step Solution

Answer

x values: \( \frac{\pi}{6}, \frac{11\pi}{6}, \frac{13\pi}{6}, \frac{23\pi}{6} \). \( y > a: x \in (0,\frac{\pi}{6}) \cup (2\pi,\frac{13\pi}{6}) \). \( y < a: x \in (\frac{\pi}{6},\frac{11\pi}{6}) \cup (\frac{13\pi}{6}, \frac{23\pi}{6}) \).

Step 1: Understand the Problem

We have the function \( y = \cos x \) on the interval \( [0, 4\pi] \). We need to find the values of \( x \) where the value of \( \cos x \) is equal to, greater than, and less than \( \frac{\sqrt{3}}{2} \).

Step 2: Finding Values of x where y = a

For \( \cos x = \frac{\sqrt{3}}{2} \), we need to identify where on the cosine graph this occurs. In the interval \( [0, 2\pi] \), \( \cos x = \frac{\sqrt{3}}{2} \) at \( x = \frac{\pi}{6} \) and \( x = \frac{11\pi}{6} \). These values repeat every full period \( 2\pi \). So in the interval \( [0, 4\pi] \), the values of \( x \) are \[ x = \frac{\pi}{6}, \frac{11\pi}{6}, \frac{13\pi}{6}, \frac{23\pi}{6} \].

Step 3: Finding Values of x where y > a

To find where \( \cos x > \frac{\sqrt{3}}{2} \), look for the portions of the cosine graph above this level. For \( \cos x \), \( \cos x > \frac{\sqrt{3}}{2} \) between \( x = 0 \) and \( x = \frac{\pi}{6} \), and symmetrically at other intervals derived from adding multiples of \( 2\pi \). This gives intervals \( (0, \frac{\pi}{6}) \), \( (2\pi, \frac{13\pi}{6}) \), plus considering the period, repeated intervals \( (2\pi, \frac{13\pi}{6}) \) and \( (4\pi, 4\pi) \).

Step 4: Finding Values of x where y < a

Now, find where \( \cos x < \frac{\sqrt{3}}{2} \). This occurs between \( x = \frac{\pi}{6} \) and \( x = \frac{11\pi}{6} \). Repeats in the other half of the graph and subsequent cycles result in intervals \( (\frac{\pi}{6}, \frac{11\pi}{6}) \), \( (\frac{13\pi}{6}, \frac{23\pi}{6}) \).

Key Concepts

Cosine Function

The cosine function, represented as \( \cos x \), is one of the fundamental trigonometric functions. It is commonly used to describe waves and circular motions. The graph of the cosine function is a smooth, continuous wave that oscillates between -1 and 1. This wave-like pattern repeats every \(2\pi\) radians, known as the period of the cosine function.

The cosine function is defined for any real number, but it is periodic, which means it repeats its values over regular intervals. These intervals help identify specific values of \(x\) where the cosine takes on particular values. In standard unit circle coordinates, \( \cos x \) represents the x-coordinate of a point on the circle.

The cosine of zero degrees or radians (\( \cos 0 \)) gives a maximum value of 1. It decreases to -1 as \(x\) approaches \(\pi\) and returns to 1 by the time it reaches \(2\pi\). This pattern allows for the use of the cosine function in solving trigonometric problems, particularly those involving equalities, inequalities, and interval-based questions.

Interval Analysis

Interval analysis is a mathematical technique used to determine the range of input (in this case, \(x\) values) that produces specific output values for a function. For the cosine function, it is crucial to understand how the function behaves over different parts of its period.

In the specified interval \([0, 4\pi]\), the cosine function covers two full periods. By analyzing the function over these increments, we can identify repeated patterns and specific sections where the function holds certain properties, such as being equal to, greater than, or less than a given value.

• When \( \cos x = \frac{\sqrt{3}}{2} \), it occurs at precise points within the interval. These solutions repeat every \(2\pi\), leading to solutions like \(x = \frac{\pi}{6}, \frac{11\pi}{6}, \frac{13\pi}{6}, \frac{23\pi}{6}\).
• Identifying intervals where \(\cos x > \frac{\sqrt{3}}{2}\) involves checking sections where the graph is above this level. These are open intervals (not including end points) derived from the given graph.
• For \(\cos x < \frac{\sqrt{3}}{2}\), we look for intervals where the cosine value dips below \(\frac{\sqrt{3}}{2}\). These are also open intervals, calculated by finding the regions between known equalities.

Understanding interval analysis aids in accurately resolving inequalities and equalities in trigonometric problems.

Inequalities in Trigonometry

Inequalities in trigonometry involve determining the range of values over which a trigonometric function is greater or less than a given quantity. This requires an understanding of the function’s behavior over its specific interval.

When addressing inequalities like \(\cos x > \frac{\sqrt{3}}{2}\) or \(\cos x < \frac{\sqrt{3}}{2}\), the cosine function's periodicity and oscillating nature are essential features. By using the known values within an interval, such as \([0, 4\pi]\), you can determine where the cosine function is above or below a certain level.

• Greater than Inequality: For \(\cos x > \frac{\sqrt{3}}{2}\), intervals like \((0, \frac{\pi}{6})\) are considered where the cosine is above the specified level.
• Less than Inequality: For \(\cos x < \frac{\sqrt{3}}{2}\), we evaluate intervals such as \((\frac{\pi}{6}, \frac{11\pi}{6})\) where the cosine value drops below \(\frac{\sqrt{3}}{2}\).

Addressing inequalities helps to understand the function's graphical representation and its practical implications. It's a valuable technique for solving problems involving trigonometric graphs, ensuring students grasp the range a trigonometric function can cover in a specified domain.