In Exercises 127–130, solve each equation on the interval [0, 2𝝅) by first rewriting the equation in terms of sines or cosines. csc² x + csc x - 2 = 0

Verified step by step guidance

1. Recognize that the equation involves the cosecant function, $\csc x$, which is the reciprocal of the sine function: $\csc x=\frac{1}{\sin x}$.
2. Substitute $\csc x=\frac{1}{\sin x}$ into the equation to rewrite it in terms of sine: ${\left(\frac{1}{\sin x}\right)}^2+\frac{1}{\sin x}-2=0$.
3. Multiply through by ${\sin }^{2}x$ to eliminate the fractions: $1+\sin x-2{\sin }^{2}x=0$.
4. Rearrange the equation to form a quadratic in terms of $\sin x$: $2{\sin }^{2}x-\sin x-1=0$.
5. Factor the quadratic equation: $(2\sin x+1)(\sin x-1)=0$, and solve for $\sin x$ to find the solutions within the interval $[0,2\pi )$.

Key Concepts

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function. It is defined as csc(x) = 1/sin(x). Understanding this function is crucial for rewriting equations involving csc in terms of sine, which simplifies the problem-solving process.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable where both sides are defined. Key identities, such as the Pythagorean identity sin²(x) + cos²(x) = 1, help in transforming equations into more manageable forms, facilitating the solution of trigonometric equations.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [0, 2π) indicates that the solutions to the equation should be found within this range, including 0 but excluding 2π. This is important for determining valid solutions in trigonometric equations, as trigonometric functions are periodic.