Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the equation

The given equation is \( \cos \theta + 1 = 0 \). To find when this equation is true, we need to find when \( \cos \theta = -1 \).

Step 2: Identify angles for the cosine function

The cosine function \( \cos \theta \) is equal to -1 at specific angles. In the unit circle, \( \cos \theta = -1 \) at \( \theta = \pi \) and also at angles coterminal with \( \pi \).

Step 3: General solution for the angle

The general solution for \( \theta \) when \( \cos \theta = -1 \) is given by \( \theta = \pi + 2k\pi \), where \( k \) is any integer. This formula accounts for all coterminal angles.

Step 4: Conclusion

Thus, the solutions to the equation \( \cos \theta + 1 = 0 \) are given by \( \theta = \pi + 2k\pi \), where \( k \) is any integer. This represents an infinite number of angles that satisfy the equation.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a fundamental trigonometric function. It helps in understanding the relationship between the angle \( \theta \) and the x-coordinate on the unit circle. The cosine function outputs values ranging from -1 to 1, based on the angle measured in radians. It is a periodic function, meaning it repeats its values every \( 2\pi \) radians.

• The cosine of 0 radians is 1.
• The cosine of \( \pi \) radians is -1.
• At angles like \( \pi/2 \) and \( 3\pi/2 \), cosine values are 0.

It is particularly useful in understanding wave patterns and oscillations. For the equation given, we look for where the cosine value equals -1, often pointing to specific angles like \( \pi \). Understanding these properties makes it easier to solve cosine-based equations.

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate system. Angles on the unit circle are measured in radians, and this circle aids in visualizing trigonometric functions like sine and cosine.

• Each point on the unit circle is defined by coordinates (cosines, sines).
• A full rotation around the circle is \( 2\pi \) radians.
• Key angle values such as \( \pi \), \( \pi/2 \), \( 3\pi/2 \), and \( 2\pi \) help locate point locations corresponding to the cosine and sine values.

Specifically, in our case, \( \cos \theta = -1 \) at \( \theta = \pi \). Angles like this can also be thought of in terms of their "coterminal" angles, as they share the same cosine and sine values when located in their respective places on the unit circle.

General Solution

The general solution in trigonometric equations refers to expressions that encapsulate all possible solutions for a given function, owing to their periodic nature. For example, when \( \cos \theta = -1 \), the angle \( \pi \) is a solution. To encompass all solutions including those derived from continuing rotations, we use the general solution formula \( \theta = \pi + 2k\pi \), where \( k \) is any integer.

• This formula accounts for every complete rotation \( 2\pi \) around the unit circle beyond the initial solution \( \pi \).
• It ensures that any angle with cosine value -1 can be expressed this way.
• The integers \( k \) can be negative, zero, or positive, illustrating backward and forward rotations on the unit circle.

This formulation is key for solving periodic equations, effectively giving us an infinite set of angle solutions for problems involving trigonometric functions.