Problem 83 - Precalculus: Functions and Graphs

Question

Approximate the solution to each inequality on the interval \([0,2 \pi]\). $$\cos x \geq 0.3$$

Step-by-Step Solution

Answer

The solutions are in the intervals \([0, 1.2661] \) and \([5.0171, 2\pi]\).

Step 1: Set up the inequality

We begin with the inequality \( \cos x \geq 0.3 \). Our goal is to find the values of \( x \) within the interval \([0, 2\pi]\) that satisfy this inequality.

Step 2: Find the reference angle

To find the reference angle, calculate \( \cos^{-1}(0.3) \). This will give us the first angle in the first quadrant that satisfies \( \cos x = 0.3 \). Let's compute it: \( \cos^{-1}(0.3) \approx 1.2661 \) radians.

Step 3: Determine the solution intervals

Since \( \cos x \geq 0.3 \), this condition holds whenever \( x \) is within the intervals where cosine values are positive. In the unit circle, these areas correspond to angles in the first and fourth quadrants. Therefore, the solution intervals are from \(0\) to \(1.2661\) and from \(2\pi - 1.2661\) to \(2\pi\).

Step 4: Calculate the endpoints

Calculate \(2\pi - 1.2661 \approx 5.0171\). Therefore, the two solution intervals where \(\cos x \geq 0.3\) are \([0, 1.2661] \) and \([5.0171, 2\pi] \).

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In mathematical terms, it is represented as \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \). The cosine function is periodic, meaning it repeats its values in regular intervals along the unit circle.
The function has a range between -1 and 1, and it is often used in solving trigonometric equations and inequalities.
For angles provided in radians, the cosine value can be determined using either trigonometric tables, calculators, or software that supports trigonometric functions. The cosine value reflects where the angle falls, such as in which quadrant it resides, as well as the behavior of the graph of the cosine function itself. Understanding cosine plays a critical role in grasping more complicated trigonometric ideas.

Reference Angle

The concept of a reference angle is essential when solving trigonometric inequalities. A reference angle is the smallest angle between the terminal side of a given angle and the horizontal axis. It is always a positive angle, measured in the range of 0 to \( \pi/2 \) radians (or 0 to 90 degrees).
Reference angles make it easier to understand and calculate the actual trigonometric values of larger angles.

• For angles in the first quadrant, the reference angle is the same as the angle itself.
• For angles in the second quadrant, the reference angle is found by subtracting the angle from \( \pi \).
• In the third quadrant, subtract \( \pi \) from the angle to find the reference angle.
• Finally, for the fourth quadrant, subtract the angle from \( 2\pi \).

The use of reference angles in the exercise simplifies the process of locating where the cosine function is positive and helps define the regions of solution within the interval \([0, 2\pi]\).

Unit Circle

The unit circle is a fundamental tool in trigonometry, representing angles and their corresponding cosine and sine values on a circle with a radius of one unit. Centered at the origin of a coordinate plane, the unit circle provides a visual and practical way to understand trigonometric functions.
Each point on the unit circle is associated with an angle, \( \theta \), measured from the positive x-axis. The x-coordinate of any point is \( \cos \theta \), while the y-coordinate is \( \sin \theta \).
This circle simplifies finding solutions to trigonometric inequalities because it illustrates how the cosine and sine functions behave as angles vary:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, and cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, and cosine is positive.

In the problem above, the unit circle aids in identifying the intervals where the cosine function maintains values greater than or equal to 0.3, particularly focusing on the first and fourth quadrants where positive cosine values exist.