Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \cos t=-1 $$

Step-by-Step Solution

Answer

The solution is \( t = \pi + 2k\pi \) where \( k \) is any integer.

Step 1: Understand the Cosine Function

The cosine function, denoted as \( \cos t \), is an even periodic function with a period of \(2\pi\), and it takes values ranging from -1 to 1 for angles \(t\) measured in radians.

Step 2: Determine Where \(\cos t = -1\)

The cosine function equals -1 at specific points. Given the periodic nature of the cosine function, it is known that \( \cos(\pi) = -1 \).

Step 3: Utilize Periodicity of the Cosine Function

Since cosine has a period of \(2\pi\), the equation \( \cos t = -1 \) will repeat at every \(2\pi\) interval. Therefore, the solutions can be expressed as \( t = \pi + 2k\pi \), where \( k \) is any integer.

Step 4: Write the General Solution

The general solution to the equation \( \cos t = -1 \) is \( t = \pi + 2k\pi \), with \( k \in \mathbb{Z} \) (any integer).

Key Concepts

Cosine Function

The cosine function, often expressed as \( \cos(t) \), is a fundamental trigonometric function. It describes the relationship between a right triangle's angle and the ratio of its adjacent side to the hypotenuse.

The cosine function is defined on the unit circle, where the \( x \)-coordinate of a point on the circle represents the cosine of the angle \( t \). Mathematically:

• The range of \( \cos(t) \) is from -1 to 1.
• The cosine of 0 degrees (or 0 radians) is 1.
• The cosine function is an even function, satisfying \( \cos(-t) = \cos(t) \).

This symmetry about the y-axis plays a crucial role in solving trigonometric equations. By understanding these properties, one can solve various trigonometric problems involving cosine.

Periodic Function

A periodic function is one that repeats its values at regular intervals. For the cosine function, this is a fundamental property.

The cosine function has a period of \( 2\pi \), meaning that every \( 2\pi \) radians, the function values repeat. We express this concept as:

• \( \cos(t + 2\pi) = \cos(t) \)
• \( \cos(t) = \cos(t + 2k\pi) \) for any integer \( k \)

This periodic nature makes it possible to predict the values of the cosine function for any angle, allowing us to find solutions over multiple cycles of the function.

The periodic property of cosine simplifies solving equations by reducing complex problems to simpler, repeating solutions.

General Solution

The concept of a general solution is crucial in trigonometry. It describes all possible solutions to a trigonometric equation.

For \( \cos(t) = -1 \), we know from the cosine function's behavior that this occurs specifically at \( t = \pi \). Given its periodic nature, the general solution incorporates the period \( 2\pi \) to account for all repetitions:

• The general solution can be written as \( t = \pi + 2k\pi \), where \( k \) is an integer.
• This expression captures every instance where \( \cos(t) = -1 \) occurs over multiple cycles.

Such solutions are integral in understanding the behavior of trigonometric functions across different contexts and conditions.

Radian Measure

Radian measure is a method of measuring angles based on the length of the arc formed by the angle on the unit circle. It is the standard measure in mathematics, especially in trigonometry.

One radian is defined as the angle formed when the arc length is equal to the radius of the circle. In terms of conversion:

• \( 2\pi \) radians equal 360 degrees.
• \( \pi \) radians equal 180 degrees.
• 1 radian is approximately 57.2958 degrees.

Understanding radians is essential when solving trigonometric equations, as it simplifies integration and differentiation processes. Utilizing radian measure allows for a seamless transition between geometry and algebra, enhancing problem-solving in trigonometry.