Problem 32 - Contemporary Precalculus

Question

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) $$\cos t=-1 / 4$$

Step-by-Step Solution

Answer

Answer: There are two solutions to the equation within the specified range.

Step 1: Recall the properties of the cosine function

The cosine function is periodic, meaning that it repeats its values in regular intervals. For the cosine function, the period is \(2\pi\). This means that, within the range from 0 to \(2\pi\), the graph of the cosine function covers all possible values between -1 and 1 exactly once.

Step 2: Sketch the cosine function

To determine visually the number of solutions to the equation, we need to sketch the graph of the cosine function, for \(t\) in the range \([0, 2\pi]\). The graph starts at its maximum value, 1, at \(t=0\), and decreases to its minimum value, -1, at \(t=\pi\). After that, it increases again to reach its maximum value at \(t=2\pi\).

Step 3: Draw the horizontal line

Next, we draw a horizontal line at \(y =-\frac{1}{4}\) in order to identify the points where the cosine function crosses the line, as these points are the solutions to the equation.

Step 4: Count the number of intersections

Visually examine the graph to determine the number of times the cosine function crosses the horizontal line \(y=-\frac{1}{4}\). There should be two intersections, one in the first half of the interval (from 0 to \(\pi\)) and one in the second half (from \(\pi\) to \(2\pi\)).

Step 5: Conclude the number of solutions

As there are two intersections between the graph of the cosine function and the horizontal line \(y=-\frac{1}{4}\) within the interval \([0, 2\pi]\), there are two solutions to the equation \(cos(t) = -\frac{1}{4}\) in this range.

Key Concepts

Cosine Function

The cosine function, written as \( \cos(t) \), is a fundamental concept in trigonometry and an essential part of the toolkit for solving many mathematical problems. It relates the angle \( t \) within a right-angled triangle to the ratio of the adjacent side to the hypotenuse. What this really means is, for any angle on a unit circle, the \( x \) coordinate at that angle represents the value of the cosine function at that angle.

When graphed, the cosine function creates a wave-like pattern, known as a sinusoidal curve, that starts from the maximum value of 1 when \( t = 0 \) and oscillates between 1 and -1. It attains its peak value of 1 at every \(2n\pi\), where \( n \) is an integer. This property is due to the cosine function's even symmetry, meaning it reflects across the y-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as the cosine function, helps us to visualize how the function behaves over different intervals. To graph \( \cos(t) \), we begin by plotting a series of points that correspond to notable angles—typically at \(0, \, \frac{\pi}{2}, \pi, \, \frac{3\pi}{2},\) and \(2 \pi\)—and their respective cosine values.

• At \( t = 0 \) and \( t = 2 \pi \), \( \cos(t) = 1 \)
• At \( t = \frac{\pi}{2} \) and \( t = \frac{3\pi}{2} \) , \( \cos(t) = 0 \)
• At \( t = \pi \), \( \cos(t) = -1 \)

Connecting these points produces the characteristic 'wave' that is typical of the function, with peaks at \(1\) and troughs at \( -1\). This graph is a visual tool in finding the solutions to equations like \( \cos(t) = -\frac{1}{4} \) by observing where the graph intersects with the horizontal line \( y = -\frac{1}{4} \).

Periodicity of Trigonometric Functions

Periodicity is a property that allows trigonometric functions to repeat their values at regular intervals. The cosine function specifically has a period of \( 2 \pi \), meaning after every interval of \( 2 \pi \), the function values start repeating. This characteristic is immensely helpful when solving trigonometric equations, as it allows one to focus on a single period to find solutions.

Understanding periodicity can be critical for tackling trigonometric problems efficiently. When given the equation \( \cos(t) = -\frac{1}{4} \) and asked to find solutions between 0 and \( 2 \pi \), we can take advantage of the cosine's periodicity, restricting our search to this interval. Since cosine is continuous and varies smoothly between its maximum and minimum values, we only expect it to reach any particular value between these extremes a limited number of times within one period, in this case, exactly two times.