Problem 19 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$2 \cos t+1=0$$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{2\pi}{3} + 2n\pi \) and \( t = \frac{4\pi}{3} + 2n\pi \), where \( n \) is an integer.

Step 1: Isolate the Cosine Function

Begin by isolating the cosine function in the equation. Start with the equation:\[ 2 \cos t + 1 = 0 \]Subtract 1 from both sides:\[ 2 \cos t = -1 \]

Step 2: Solve for Cosine

To solve for \( \cos t \), divide both sides of the equation by 2:\[ \cos t = -\frac{1}{2} \]

Step 3: Find the General Solution

The general solution for \( \cos t = -\frac{1}{2} \) can be found using the unit circle. Recall that the cosine of an angle is \(-\frac{1}{2}\) at angles \( t = \frac{2\pi}{3} + 2n\pi \) and \( t = \frac{4\pi}{3} + 2n\pi \), where \( n \) is any integer.

Key Concepts

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle centered at the origin of a coordinate system with a radius of 1.
The circumference of this circle is expressed in radians, typically ranging from 0 to \(2\pi\). In the context of trigonometric functions, the unit circle provides a geometric way to study the behavior of the sine, cosine, and tangent functions.

• Each point on the unit circle corresponds to an angle \( t \), measured in radians.
• The x-coordinate of a point on the unit circle gives the value of \( \cos t \).
• The y-coordinate of a point gives the value of \( \sin t \).

For example, the angle \( \frac{2\pi}{3} \) has a cosine value of \(-\frac{1}{2}\), and similarly, \( \frac{4\pi}{3} \) gives the same cosine value. These points are symmetrically located around the vertical axis of the circle.

General Solution

In trigonometry, finding a general solution involves identifying all possible angles that satisfy a given trigonometric equation.
This typically goes beyond finding just one solution, as angles that differ by full rotations \(2\pi\) from each other still yield the same trigonometric value.
When you find a particular solution, adding \(2n\pi\) - where \(n\) is an integer - allows you to express all possible angles that solve the equation.

• In the example \( \cos t = -\frac{1}{2} \), we identify
  • \( t = \frac{2\pi}{3} + 2n\pi \)
  • \( t = \frac{4\pi}{3} + 2n\pi \)
  as the general solutions.
• This effectively captures all the unique rotations, ensuring no solution is missed within the context of periodicity.

Cosine Function

The cosine function is one of the primary functions in trigonometry and is fundamentally linked to the unit circle.
Represented as \( \cos t \), it describes the horizontal coordinate of an angle \( t \) on the unit circle.

• The function is periodic with a period of \(2\pi\), meaning \( \cos(t + 2\pi) = \cos t \).
• Cosine values range between \(-1\) and \(1\), correlating with angles on the unit circle.
• Positive values indicate positions on the right side, while negative values indicate the left side of the unit circle.

To solve equations like \( \cos t = -\frac{1}{2} \), it is crucial to understand where these values occur on the unit circle.

• These specific angles, such as \( \frac{2\pi}{3} \) and \( \frac{4\pi}{3} \), are key because they reflect where the x-coordinates (cosine values) are \(-\frac{1}{2}\).

Understanding these positions helps extract all relevant solutions and navigate trigonometric equations effectively.