Problem 31 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\cos \frac{x}{2}-1=0$$

Step-by-Step Solution

Answer

The solutions are of the form \( x = 4k\pi \) where \( k \) is any integer.

Step 1: Simplify the equation

Start by rewriting the given equation. We have \( \cos \frac{x}{2} - 1 = 0 \). To solve for \( x \), add 1 to both sides: \( \cos \frac{x}{2} = 1 \).

Step 2: Identify solutions with cosine function

The cosine function equals 1 at specific angles, which are multiples of the full circle, \( 0, 2\pi, 4\pi, \ldots \). This can be represented as \( \frac{x}{2} = 2k\pi \), where \( k \) is any integer.

Step 3: Solve for x

Multiply both sides of the equation by 2 to solve for \( x \). This gives us \( x = 4k\pi \). Hence, any angle that is a multiple of \( 4\pi \) satisfies the original equation.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, crucial for solving trigonometric equations. It relates the angle in a right-angled triangle to the ratio of the adjacent side to the hypotenuse. In this context, its value can tell us about the angle's position on the unit circle.

The function is periodic with a period of \(2\pi\), meaning the values repeat every full circle (or \(360^\circ\)). When the cosine of an angle is \(1\), it corresponds to a point on the unit circle where the angle is aligned with the positive x-axis.

• Cosine equals 1 at \(0, 2\pi, 4\pi, \ldots\)
• These are the "resting" points at a full rotation, making it a key property in solving trig equations

Simplifying and solving these equations often involves understanding the basic characteristics of the cosine curve and knowing when it reaches specific values like 1.

Angle Multiples

When solving trigonometric equations involving angle multiples, it is essential to grasp how angles work within the constraints of periodic functions like the cosine. In our equation, \( \cos \frac{x}{2} = 1\), the division within the angle suggests that we are working with half-angle multiples.

In this equation, \( \frac{x}{2}\) implies that the angle x is twice as large when considering multiples for the solution. Multiplying \( \frac{x}{2} = 2k\pi \) by 2 ensures the original angle x corresponds to these specific points on the unit circle where the cosine (of half-angle) equals 1.

• Understanding angle multiples means associating angle changes with predictable positions on the circle.
• Half-angle expressions require multiplying to revert to the standard angle.

This concept is key when dealing with transformations in trigonometric identities and equations.

General Solutions

General solutions to trigonometric equations allow us to express all possible solutions using expressions involving the angle and a constant term. In this context, \(x = 4k\pi\) denotes that every solution is a multiple of \(4\pi\). These solutions cover the infinite series of possibilities due to the periodic nature of trigonometric functions.

By introducing the integer \(k\), we capture every instance where the original equation holds true as x wraps around the circle entirely.

• General solutions harness periodicity to express every valid x.
• They provide a comprehensive way to capture all angles satisfying the equation.

For students, understanding general solutions ensures not only finding specific answers but generalizing for any scenario.