Problem 44 - 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 of \(x\) are \(\pi\) and \(3\pi\).

Step 1: Understand the Problem

We need to find the values of \(x\) within the interval \([0, 4\pi]\) that satisfy the equation \(\cos x = -1\). Here, we are dealing with the cosine function, which has a range of [-1, 1].

Step 2: Identify the Property of Cosine

The cosine function equals -1 at specific points. The fundamental property of the cosine function is that \(\cos x = -1\) at integer multiples of \(\pi\), specifically at odd multiples, because its period is \(2\pi\).

Step 3: Determine the Specific Values of x

To find when \(\cos x = -1\), we can use the known values from the unit circle where cosine equals -1, which is at odd multiples of \(\pi\). Within the interval \([0, 4\pi]\), identify these values.

Step 4: Calculate Solutions

The cosine function equals -1 at \(x = \pi, 3\pi, 5\pi, \, \ldots\). However, since we are restricted to \([0, 4\pi]\), we only consider \(x = \pi\) and \(x = 3\pi\).

Step 5: Verify Solutions

Verify that the calculated solutions satisfy the original equation \(\cos x = -1\). We check:\[\cos(\pi) = -1 \quad \text{and} \quad \cos(3\pi) = -1\]Hence, both values are correct.

Key Concepts

Cosine Function

In trigonometry, the cosine function is one of the primary functions used to understand angles and lengths in triangles. It relates an angle in a right triangle to the length of the adjacent side over the hypotenuse. The cosine function is periodic and oscillates between -1 and 1. This periodicity means it repeats its pattern every 2π radians.

• The cosine of an angle \(x\) can be visualized on a graph, with peaks and troughs at \(y = 1\) and \(y = -1\) respectively.
• Specifically, the cosine function equals -1 at specific points, namely at odd multiples of \(\pi\), such as \(x = \pi \) or \(x = \3\pi\).

Understanding when and where the cosine function reaches these values is pivotal in solving trigonometric equations such as \(\cos x = -1\). By knowing the behavior of the cosine function, you can pinpoint where these occurrences will take place within the given interval.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle is instrumental in defining trigonometric functions for all real numbers. Every point on the unit circle corresponds to an angle formed with the positive x-axis. The x-coordinate represents the cosine of the angle, whereas the y-coordinate represents the sine. Thus, to find \(\cos x = -1\), one would look for the point(s) on the unit circle where the x-coordinate equals -1.

• The cosine of an angle on the unit circle equals -1 at \(x = \pi \) and \(x = \3\pi\) within the \( \[0, 4\pi \] \) interval.
• These values are significant because they form the boundaries of one full cycle on the unit circle.With the unit circle, these solutions are readily visualized and understood in the context of a rotating angle around the circle.

Being adept with the unit circle equips students with a powerful method of visualizing and solving trigonometric problems.

Interval Analysis

When solving trigonometric equations, interval analysis plays a crucial role. This process involves finding solutions to equations within specified bounds, such as \(\[0, 4\pi\] \). Understanding the behavior of trigonometric functions within these intervals helps identify relevant solutions.

• The interval \(\[0, 4\pi\] \) represents two full cycles of the cosine function, as its period is \(2\pi\).
• By narrowing our focus to specific intervals, we can ensure that we capture all possible solutions without overextending beyond the required range.

Interval analysis is particularly helpful for avoiding unnecessary calculations. For instance, when determining where \(\cos x = -1\), restricting our search to specific intervals ensures efficient problem-solving. This method is consistently applied in mathematics to solve various equations.

Trigonometric Identities

Trigonometric identities are essential equations involving trigonometric functions that are true for all allowed values of the involved angles. They are instrumental in deriving solutions, simplifying calculations, and proving other mathematical concepts.Common identities include:

• Pythagorean Identity: \(\\sin^2 x + \\cos^2 x = 1\)
• Double Angle Formulas: E.g., \(\\cos(2x) = 2\cos^2 x - 1\)
• Angle Sum and Difference Formulas: \(\\cos(x \pm y) = \\cos x\cos y \mp \\sin x\sin y\)

For instance, knowing the cosine double angle formula can help understand the periodicity and symmetry of the function.By mastering these identities, students are better equipped to manipulate, simplify, and solve trigonometric equations efficiently. They provide a framework from which more complex concepts can be understood and applied.