Problem 7 - Precalculus with Calculus Previews

Question

In Problems \(7-12\), find all solutions of the given trigonometric equation if \(x\) represents a real number. $$ \cos x=-1 $$

Step-by-Step Solution

Answer

\(x = \pi + 2k\pi\), where \(k\) is an integer.

Step 1: Understand the Unit Circle

In trigonometry, the cosine of an angle corresponds to the x-coordinate of the point on the unit circle. Since \(\cos x = -1\), we are looking for the angle where the x-coordinate is -1 on this circle.

Step 2: Identify the Point on the Unit Circle

On the unit circle, the point where the cosine value is -1 is located at \(x = \pi\). This is because at this angle, the x-coordinate is -1 while the y-coordinate is 0.

Step 3: Consider the Periodicity of Cosine

The cosine function is periodic with a period of \(2\pi\). This means that if \(x = \pi\) is a solution, then ... \(x = \pi + 2k\pi\) for any integer \(k\) is also a solution.

Step 4: Write Down the General Solution

Thus, the general solution for the equation \(\cos x = -1\) is given by \[x = \pi + 2k\pi\]\ where \(k\) is any integer.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand the properties of trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.

In this circle, any angle corresponds to a point on the circumference where:

• The x-coordinate of the point represents the cosine of that angle.
• The y-coordinate represents the sine of the angle.

For example, if we are given an equation such as \(\cos x = -1\), we need to find the angle on the unit circle where the x-coordinate is -1.

This occurs precisely at the point \((-1, 0)\), which corresponds to the angle \(x = \pi\). Understanding how angles relate to points on the unit circle is crucial for solving trigonometric equations.

Cosine Function

The cosine function is one of the primary trigonometric functions, symbolized as \(\cos\). It relates an angle to the x-coordinate of the corresponding point on the unit circle.

The cosine function has a few key properties:

• It is an even function, meaning that \(\cos(-x) = \cos(x)\).
• Its range is between -1 and 1 inclusive.
• It attains the value of -1 at specific angles, such as \(\pi\), where the x-coordinate on the unit circle is -1.

To solve the equation \(\cos x = -1\), we find the angle where the x-coordinate on the unit circle equals -1, which occurs at \(x = \pi\). This identification helps us recognize solutions in trigonometric equations.

Periodicity of Trigonometric Functions

Trigonometric functions, like the cosine function, are periodic. This means they repeat their values in regular intervals. Understanding this periodic nature is essential when solving equations.

For the cosine function, the period is \(2\pi\). Hence, if we have an angle \(x = \pi\) as a solution to the equation \(\cos x = -1\), we acknowledge that:

• Cyclic results occur every \(2\pi\) units.
• The general form for all solutions to this equation is \(x = \pi + 2k\pi\), where \(k\) is any integer.

This periodicity helps us extend our single solution into an infinite set of solutions, capturing the essence of trigonometry's repetitive nature.