Problem 2 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$\cos t=-1$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Unit Circle

The equation \( \cos t = -1 \\) refers to the unit circle properties. The most crucial fact to remember is that the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Step 2: Recall the Key Angle

On the unit circle, the cosine value of -1 occurs where the angle is directed towards the negative x-axis. This position is precisely at \( \pi \\) radians (or 180° ).

Step 3: General Solution for Cosine Function

The cosine function is periodic with a period of \( 2\pi \\). Therefore, if \( t = \pi \\) is a solution, then all additional solutions take the form \( t = \pi + 2k\pi \\), where \( k \\) is any integer. This accounts for the periodic nature of the cosine function.

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry that provides a simple yet powerful way to understand angles and trigonometric functions. Imagine a circle with a radius of 1 that is centered at the origin of a coordinate plane. Every point on this circle can be represented as \( (\cos \theta, \sin \theta) \), where \theta\ is the angle measured in radians from the positive x-axis.

• The x-coordinate corresponds to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.

When you encounter a problem like \( \cos t = -1 \), the unit circle helps you visualize where this condition is true. For \cos t = -1\, you look for a point where the angle's terminal side intersects the unit circle's circle on the negative x-axis. This occurs at the angle of \pi \ radians (or 180°), giving us a clear visual tool for solving trigonometric equations.

Cosine Function

The cosine function is a key component of trigonometry that pairs well with the unit circle. It is defined as the adjacent side over the hypotenuse in a right triangle, but it naturally extends to work within the context of the unit circle.

• The range of cosine is \[-1, 1\], meaning it outputs values anywhere from -1 to 1.
• Its graph oscillates between these values, creating a wave-like pattern.

In relation to the exercise statement, \( \cos t = -1 \) translates to finding where on the circle the x-value equals -1. This only happens once in each cycle of the cosine function, specifically at \pi \ radians. Understanding the behavior of the cosine function on the unit circle aids in determining solutions to trigonometric equations like these.

Periodic Functions

Periodic functions repeat their values at regular intervals; the cosine function is an excellent example of this behavior.

• The cosine function has a period of \2\pi\, meaning that every \2\pi\ units, the function repeats its entire shape.
• This property explains why the general solution for \cos t = -1\ involves \pi + 2k\pi\.

When a function has a period \(T\), any solution at \(t_0\) can generate other solutions in the form \(t = t_0 + nT\), where \(n\) is any integer. For \cos t = -1\, knowing the period lets us write solutions as \(t = \pi + 2k\pi\). This accounts for all possible angles that satisfy the given cosine condition repeatedly along the unit circle, demonstrating the elegant symmetry of periodic functions.