Problem 1 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\cos x+1=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \pi + 2k\pi\) for integer \(k\).

Step 1: Isolate the Trigonometric Function

Begin by isolating the cosine function in the given equation: \[ \cos x + 1 = 0 \]Subtract 1 from both sides to solve for the cosine function:\[ \cos x = -1 \]

Step 2: Determine Angle with Cosine of -1

Identify the angle(s) where the cosine value is -1. Cosine reaches -1 at specific angles on the unit circle. The primary angle is:\[ x = \pi + 2k\pi, \text{ for any integer } k \]This is because \(\cos(\pi) = -1\) and it repeats every full circle, \(2\pi\).

Step 3: List General Solutions

The general solution for the equation \(\cos x = -1\) is:\[ x = \pi + 2k\pi, \text{ where } k \text{ is any integer} \]

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, often abbreviated as "cos." It is a periodic function, which means that it repeats its values in a regular pattern over intervals. Cosine is specifically related to the x-coordinate of points on the unit circle. This function is an even function due to its symmetry about the vertical axis. With values ranging from

• -1 to 1
• cosine is particularly useful in calculating angles and lengths in right-angle triangles.

Let's consider its equation \[\cos x = -1\]to illustrate this. This equation tells us that we are looking for angles at which the x-coordinate on the unit circle is -1. Such a point is at the far left side of the circle. The cosine curve will intercept this value in a predictable manner due to its periodic nature.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The basic principle of the unit circle is that any point on it can be described using trigonometric functions. Here,

• the cosine of an angle corresponds to the x-coordinate of its point
• the sine of an angle corresponds to the y-coordinate.

Using the unit circle, students can find the cosine of common angles like 0, \(\frac{\pi}{2}\), \(\pi\), and others. The cosine angle is -1 at exactly \(\pi\) radians or 180 degrees. This is because at this point on the unit circle, the x-coordinate is -1. The periodicity of the unit circle also ensures that the same value of cosine repeats every full revolution (or \(2\pi\) radians). Hence, this angle translates to the solution of trigonometric equations like the one being tackled in the original exercise.

General Solution

In trigonometry, finding the general solution to an equation means determining all possible solutions rather than just a single instance. For equations involving cosine, once you identify one angle where the requirement is met, you can find others using its periodic nature.
For the equation \[\cos x = -1\]in the exercise, we found that \[x = \pi\]is a solution. But due to the repeating nature of cosine values, any angle that is multiple of a full circle added to \(\pi\), forms another valid solution.
This can be expressed in general terms as \[x = \pi + 2k\pi\]where \(k\) is any integer. Hence, every increment of \(2\pi\) from \(\pi\) gives another angle where the cosine of x equals -1. This representation helps students understand that trigonometric equations can have infinitely many solutions, seamlessly fitting the periodic pattern of angles generated by trigonometric functions.