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Chapter 7: Problem 67

Find all solutions to \((\sin x-1)(\sin x+1)=0\) in the interval \((0,2 \pi)\)

Short Answer

Expert verified
The solutions are \(x = \frac{\pi}{2} \) and \(x = \frac{3\pi}{2}\).

Step by step solution

01

Identify the equation factors

The given equation \((\sin x-1)(\sin x+1)=0\) can be solved by setting each factor to zero. This gives us two separate equations: \(\sin x -1 = 0\) and \(\sin x + 1 = 0 \).
02

Solve the first equation

Solve \(\sin x - 1 = 0 \). Adding 1 to both sides, we get \(\sin x = 1\). The sine of an angle is 1 at \(x = \frac{\pi}{2} \).
03

Solve the second equation

Solve \(\sin x + 1 = 0 \). Subtracting 1 from both sides, we get \(\sin x = -1 \). The sine of an angle is -1 at \(x = \frac{3\pi}{2} \).
04

Verify solutions within the interval

The solutions \(x = \frac{\pi}{2} \) and \(x = \frac{3\pi}{2} \) are within the given interval \((0, 2\pi)\). Therefore, no further adjustments are needed.

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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 \(\sin x\), describes the sine of an angle in a right-angled triangle. It represents the ratio of the length of the opposite side to the hypotenuse. This function is periodic with a period of \(2\pi\). Its values repeat every \(2\pi\), meaning \(\sin (x + 2\pi) = \sin x\). The sine function oscillates between -1 and 1, taking on every value within this range at least once in one cycle. Knowing where \(\sin x =1\) or \(\sin x =-1\) is crucial for solving trigonometric equations.
Interval Notation
In this exercise, we need to find solutions in the interval \((0, 2\pi)\). Interval notation is a way of writing subsets of the real number line. Here, \((0, 2\pi)\) indicates all the numbers between 0 and \(2\pi\), not including the endpoints. This notation is important because it specifies the range within which we need to find our solutions. When considering trigonometric functions, interpreting intervals correctly ensures we identify all relevant solutions.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved angles. The given equation \((\sin x-1)(\sin x+1)=0\) is simplified using the identity property of multiplication. For two products, \(a \cdot b=0\), either \ a \ or \ b \ must be zero. Therefore, \(\sin x - 1=0\) or \(\sin x + 1=0\) are the key components leading to the solutions. Using identities helps to break down and simplify complex equations systematically.
Zero-Product Property
The zero-product property is fundamental in algebra. It states that if the product of two factors is zero, then at least one of the factors must be zero. In the equation \((\sin x-1)(\sin x+1)=0\), the property tells us that either \(\sin x - 1 = 0\) or \(\sin x + 1 = 0\). Solving these individually gives \(\sin x = 1\) and \(\sin x = -1\). This property is critical for splitting the problem into manageable parts and then solving each part separately to find all possible solutions.

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