Problem 2 - Precalculus : Building Concepts and Connections

Question

For what value(s) of \(\theta\) in \([0,2 \pi]\) does \(\cos \theta\) reach a minimum value?

Step-by-Step Solution

Answer

\(\theta = \pi\) is the value at which the cosine function reaches its minimum value on the interval \([0, 2 \pi]\).

Step 1: Recall the properties of the cosine function

The cosine function on the interval \(0\) to \(2 \pi\) first starts at a maximum value of 1 at \(\theta = 0\), decreases to -1, then increases back to 1. The function reaches its minimum value at \(\theta = \pi\).

Step 2: Find the value of \(\theta\) giving the minimum cosine value

\(\cos(\pi) = -1\), which is the minimum value of cosine function in the interval \([0,2 \pi]\). So, \(\theta = \pi\) gives the minimum value.

Key Concepts

Cosine Function

The cosine function is a fundamental component of trigonometry. It relates the angle of a right triangle to the ratio of the lengths of the side adjacent to the angle over the hypotenuse. The cosine function is periodic, meaning it repeats its values in regular intervals. The standard period of the cosine function is \(2 \pi\). This means that every \(2 \pi\) units along the x-axis, the function returns to the same value.

In a unit circle, cosine represents the x-coordinate of a point that lies on the circle, where the circle's radius is 1. This concept helps in visualizing how the cosine function varies as an angle grows from 0 to \(2 \pi\).

The cosine function begins at 1 when the angle \(\theta\) is 0, decreases to -1 as \(\theta\) approaches \(\pi\), and then increases back to 1 by the time \(\theta\) reaches \(2 \pi\). This pattern results in a smooth, wave-like graph, which peaks at 1 and troughs at -1. This knowledge is vital when determining key points in the function, such as max and min values.

Trigonometric Properties

Trigonometric functions, like cosine, have several unique properties that allow us to solve problems involving periodic behavior. Key properties include symmetry, periodicity, and even-odd identities. These properties govern how the functions behave and can dictate their maxima and minima.

For cosine, one of its important properties is that it is an even function. This means that cosine is symmetric about the y-axis. Mathematically, this is expressed as \(\cos(-\theta) = \cos(\theta)\). This symmetry is useful to identify patterns in the cosine curve without recalculating every point.

• The cosine function's range is always from -1 to 1. This indicates that it will never exceed these bounds.
• It achieves its maximum value of 1 when the angle \(\theta\) is 0, or multiples of \(2 \pi\).
• Minimum value of -1 occurs at \(\theta = \pi\) within the interval \([0, 2\pi]\).

Understanding these attributes helps in tackling problems related to the function's behavior over specific intervals.

Interval Evaluation

Interval evaluation in trigonometry involves checking how a function behaves within a defined range of angles. For the task of finding a minimum value, it's important to understand how the cosine curve moves between these two boundaries.

In the initial problem, we look for the minimum of \(\cos \theta\) in the interval \([0, 2 \pi]\). Evaluating cosine across this interval means identifying where it hits its lowest point. We use the periodic property of the cosine function, knowing it hits -1 exactly at \(\theta = \pi\).

The symmetry properties of the cosine graph help us predict these extremities without having to compute each value individually. Because the trough at \(\pi\) is the lowest point the cosine can reach in its range, it is simple to confirm that this is the minimum value within the specified interval. Employing these methods of interval evaluation can drastically simplify solving trigonometric problems.