91Ó°ÊÓ

The sine function, denoted as \( \sin x \), is one of the primary functions in trigonometry. It's defined as the ratio of the opposite side to the hypotenuse in a right triangle. For an angle \( x \), the sine function gives the vertical coordinate of the point on the unit circle corresponding to that angle. It is a periodic function with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) units. This property is helpful when solving trigonometric equations within specific intervals. The range of the sine function is between -1 and 1, making it useful in modeling wave-like phenomena.

• Range: \([-1, 1]\)
• Period: \(2\pi\)
• Key Values: \( \sin 0 = 0 \), \( \sin \frac{\pi}{2} = 1\), \( \sin \pi = 0 \), and \( \sin \frac{3\pi}{2} = -1\)

It's important to detect when \( \sin x \) attains specific values, like 0, 1, or -1, to determine solutions to trigonometric equations on given intervals.

The cosecant function, denoted \( \csc x \), is the reciprocal of the sine function. It is defined as \( \csc x = \frac{1}{\sin x} \). Because it is a reciprocal, \( \csc x \) is undefined wherever \( \sin x = 0 \). This occurs at multiples of \( \pi \), where the sine function equals zero.The range of cosecant does not include values between -1 and 1 because these would require dividing by a number greater than 1 in absolute value, resulting in undefined or infinite values.

• Domain: All real numbers except integer multiples of \( \pi \)
• Range: \(( -\infty, -1 ] \cup [ 1, \infty ) \)

The concept of \( \csc x \) is useful in trigonometric equations involving reciprocals, like those in our exercise. By expressing \( \csc x \) as \( \frac{1}{\sin x} \), equations can be transformed to focus on \( \sin x \), making them simpler to solve.

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \). In trigonometry, we often encounter quadratic equations with trigonometric functions, as in the given problem.The equation \( \sin^2 x - 2 \sin x + 1 = 0 \) is a quadratic in terms of \( \sin x \). It can be treated like any standard quadratic equation.

• Standard form: \( a(x - p)^2 = 0 \) for perfect squares
• Solution method: Factoring, quadratic formula, or completing the square

For this equation, recognizing it as a perfect square \((\sin x - 1)^2 = 0\) makes it straightforward to solve, as \( \sin x = 1 \). Such simplifications allow us to find the values of \( x \) that satisfy the original trigonometric equation.

Interval notation is a mathematical shorthand used to describe a range of numbers along an axis. It helps clearly define the set of solutions for an equation, especially when these solutions are limited to specific intervals.In the context of trigonometric equations, interval notation specifies where solutions are valid. For instance, the interval \( 0 \leq x < 2\pi \) indicates that \( x \) includes all values from 0 up to, but not including, \( 2\pi \). There are specific symbols used in interval notation:

• Brackets \([\ ]\): Denote inclusion of endpoint (closed interval)
• Parentheses \((\ )\): Denote exclusion of endpoint (open interval)

Understanding intervals is crucial in trigonometry as most periodic functions are evaluated within a set interval, necessitating precision in which endpoints are or aren't part of the solution set.