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Chapter 7: Problem 2

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

Short Answer

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

Step by step solution

01

Understand the Equation

The given equation is \( \sin x + 1 = 0 \). Our goal is to find values of \( x \) that satisfy this equation. First, we'll isolate the sine function by subtracting 1 from both sides to get \( \sin x = -1 \).
02

Identify the Sine Function Value

Recall that the sine function \( \sin x \) reaches -1 at specific points on the unit circle. Specifically, \( \sin x = -1 \) at \( x = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer, because the sine function has a period of \( 2\pi \).
03

Define the General Solution

Using the property of periodicity of the sine function, the general solution for \( x \) given \( \sin x = -1 \) is \( x = \frac{3\pi}{2} + 2k\pi \). This equation accounts for all angles that measure -1 for \( \sin x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine Function
The sine function, denoted as \( \sin x \), is a fundamental concept in trigonometry. It describes the y-coordinate of a point on the unit circle for a given angle \( x \). When you analyze the sine function, it's important to understand that it can take values ranging from -1 to 1.

Here is how the sine function works:
  • The sine of an angle is the vertical coordinate of the corresponding point on the unit circle.
  • \( \sin(0) = 0 \), since at angle 0, the y-coordinate is 0.
  • \( \sin(\pi/2) = 1 \), at this angle, the y-coordinate reaches its maximum value (the top of the circle).
  • \( \sin(\pi) = 0 \), curving back to the x-axis.
  • \( \sin(3\pi/2) = -1 \), where the y-coordinate reaches its lowest value (the bottom of the circle).
These characteristics of the sine function are crucial when solving trigonometric equations because they set the baseline for understanding where the function crosses specific values on the circle.
Unit Circle
The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It's an essential tool in trigonometry for understanding angles and their corresponding sine and cosine values.

In the context of the unit circle:
  • The unit circle helps to visualize the trigonometric functions and their properties.
  • Each point on the unit circle corresponds to an angle measure, where the x-coordinate is the cosine and the y-coordinate is the sine of that angle.
  • For example, the point \((0, -1)\) represents \(3\pi/2\) radians, where \(\sin(3\pi/2) = -1\).
This visualization aids in understanding concepts like periodicity and helps to intuitively solve trigonometric equations. When solving \( \sin x = -1 \), the unit circle is used to identify where the sine function equals -1, helping to determine the general solution for the equation.
Periodicity
Periodicity in trigonometry refers to the repeating nature of sine, cosine, and other trigonometric functions.

Understanding periodicity involves several key points:
  • The sine function has a periodicity of \(2\pi\), meaning that the function repeats its values every \(2\pi\) interval.
  • This property is crucial for finding all solutions to trigonometric equations, as it allows us to express solutions in a recurring pattern.
  • For example, the general solution for \(\sin x = -1\) is \(x = \frac{3\pi}{2} + 2k\pi\), where \(k\) is an integer. The term \(2k\pi\) accounts for each full rotation of the circle, returning to an angle where \(\sin x\) is again -1.
Periodicity is what enables us to use one solution of a trigonometric equation to find infinitely many others, simply by adding whole periods to the angle, making this concept integral to solving equations involving trigonometric functions.

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