Problem 2 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\sin x+1=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{3\pi}{2} + 2k\pi \) where \( k \) is an integer.

Step 1: Simplify the Equation

First, let's simplify the equation by isolating \( \sin x \). We start with the equation \( \sin x + 1 = 0 \). Subtracting 1 from both sides, we get \( \sin x = -1 \).

Step 2: Identify Specific Solutions

Next, we need to find the values of \( x \) for which \( \sin x = -1 \). The sine function attains the value of \(-1\) at specific points. From trigonometric identities, we know \( \sin x = -1 \) when \( x = \frac{3\pi}{2} + 2k\pi \) where \( k \) is an integer.

Step 3: Write the General Solution

Since the sine function is periodic with a period of \(2\pi\), the general solution for \( x \) can be expressed as \( x = \frac{3\pi}{2} + 2k\pi \) where \( k \) is any integer. This accounts for all possible solutions as the sine wave repeats every \(2\pi\).

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It is often represented as \( \sin(x) \). The sine function is important as it describes the y-coordinate of a point on the unit circle—a circle with a radius of one unit.

Key properties of the sine function include its range and its period:

• Range: The values of sine repeat between -1 and 1. This means for any angle \( x \), \( \sin(x) \) will always fall within this range.
• Odd function: The sine function has symmetry around the origin, meaning \( \sin(-x) = -\sin(x) \).
• Continuous wave: It produces a smooth curve which is useful in modeling wave patterns.

These properties play a critical role in solving trigonometric equations, such as \( \sin x = -1 \). The understanding that the sine function is continuous and periodic helps identify both specific and general solutions to such equations.

Periodicity

Periodicity refers to the repeating pattern of a function over regular intervals. For the sine function, this is particularly important. The sine function has a period of \(2\pi\), which effectively means its graph repeats every \(2\pi\).

When solving equations involving periodic functions like sine, understanding the concept of periodicity is essential. This allows you to:

• Identify all solutions: Find not just a single solution but all possible values that meet the equation within one cycle and beyond.
• Define general solutions: Express solutions in terms of the period. For \( \sin x = -1 \), knowing the function's periodicity lets you express solutions as \( x = \frac{3\pi}{2} + 2k\pi \), where \( k \) is an integer.

By recognizing the repeating nature of the sine wave, you can solve complex trigonometric equations more easily.

Trigonometric Identity

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables. They are incredibly useful when simplifying trigonometric equations.

Some of the common identities include:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identities: These identities express \( \sin(x \pm y) \) and \( \cos(x \pm y) \) in terms of sin and cos individual angles.
• Double and Half Angle Identities: These help break down angles into more manageable parts or combine them.

In the context of the problem \( \sin x = -1 \), the identity that is particularly useful is understanding the sine function's value at specific points on the unit circle like \( \sin(\frac{3\pi}{2}) = -1 \), which repeats across each period.

Utilizing these identities is crucial for a deeper understanding of trigonometric functions and solving equations using them.