Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-46: Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$ \cos x=1 $$

Step-by-Step Solution

Answer

The values are \(x = 0\), \(x = 2\pi\), and \(x = 4\pi\).

Step 1: Understanding the Problem

We need to find all values of \(x\) in the interval \([0,4\pi]\) for which \(\cos x = 1\). The equation involves the cosine function, and the interval specifies the range of possible solutions.

Step 2: Recognizing Cosine Characteristics

The cosine function \(\cos x\) reaches its maximum value of 1 when \(x = 0, 2\pi, 4\pi, \ldots\). This is due to the periodicity of the cosine function, which repeats every \(2\pi\).

Step 3: Finding Solutions in Specified Interval

Within the interval \([0, 4\pi]\), we identify the specific points where \(\cos x = 1\). These points occur at \(x = 0\), \(x = 2\pi\), and \(x = 4\pi\).

Step 4: Listing the Values

The exact values of \(x\) in the interval \([0, 4\pi]\) that satisfy \(\cos x = 1\) are precisely \(x = 0\), \(x = 2\pi\), and \(x = 4\pi\).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It represents the x-coordinate of a point on the unit circle as the circle is rotated by an angle \( x \) from the positive x-axis. The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. The equation \( \cos x = 1 \) involves finding when the x-coordinate of a point on the unit circle is exactly 1. This only happens when the point is directly on the positive x-axis, which simplifies finding solutions within specified intervals. Key Properties:

• Values range from -1 to 1.
• It is an even function, meaning \( \cos(-x) = \cos(x) \).
• Reaches maximum value 1 at multiples of \( 2\pi \) (e.g., \( x = 0, 2\pi, 4\pi, \ldots \)).

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals, known as periods. Trigonometric functions like sine and cosine are classic examples of periodic functions.For the cosine function:

• The standard period is \( 2\pi \).
• This means \( \cos(x) = \cos(x + 2k\pi) \) for any integer \( k \).

In the context of solving \( \cos x = 1 \), the periodic nature is crucial because it helps identify all possible solutions within a given range by considering shifts of \( 2\pi \). Each time the function goes through its complete cycle, it returns to the starting value, such as when \( x = 0 \) and \( x = 2\pi \), showing how periodicity aids in finding multiple solutions.

Unit Circle

The unit circle is a powerful tool for understanding trigonometric functions. It is a circle with radius 1 centered at the origin (0,0) on a coordinate plane. Defining Trigonometric Functions:

• The x-coordinate of a point on the unit circle is \( \cos(x) \), reflecting the cosine function.
• The y-coordinate represents \( \sin(x) \), the sine function.

The unit circle is especially helpful for visualizing when trigonometric functions reach their maximum, minimum, or specific values like 1. Recalling how \( \cos x \) equals 1 highlights when the point on the unit circle aligns exactly on the positive x-axis, at angles \( 0, 2\pi, \) and \( 4\pi \) within the given range.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables they include, involving trigonometric functions like sine, cosine, and tangent. These identities are crucial for simplifying expressions and solving trigonometric equations. Common Identities:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Even-Odd Identities: \( \cos(-x) = \cos(x) \)
• Angle Sum Identities: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)

In the exercise of \( \cos x = 1 \), understanding these identities reinforces how and why cosine hits specific values like 1, particularly its use in confirming that solutions like \( x = 0, 2\pi, \) and \( 4\pi \) align well with the known properties of cosine.