In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 10 cos² x + 3 sin x - 9 = 0

Verified step by step guidance

1. Step 1: Use the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$ to express ${\cos }^{2}x$ in terms of $\sin x$. This gives ${\cos }^{2}x=1-{\sin }^{2}x$.
2. Step 2: Substitute ${\cos }^{2}x=1-{\sin }^{2}x$ into the equation, resulting in $10(1-{\sin }^{2}x)+3\sin x-9=0$.
3. Step 3: Simplify the equation to $10-10{\sin }^{2}x+3\sin x-9=0$, which further simplifies to $-10{\sin }^{2}x+3\sin x+1=0$.
4. Step 4: Rearrange the equation to form a quadratic equation in terms of $\sin x$: $10{\sin }^{2}x-3\sin x-1=0$.
5. Step 5: Solve the quadratic equation $10{\sin }^{2}x-3\sin x-1=0$ using the quadratic formula $\sin x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=10$, $b=-3$, and $c=-1$.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity (sin²x + cos²x = 1) and the double angle formulas. Understanding these identities is crucial for simplifying trigonometric equations and transforming them into a solvable form.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants. In the context of trigonometric equations, we often encounter quadratic forms in terms of sin x or cos x. Recognizing how to factor or apply the quadratic formula is essential for finding the solutions to these equations.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this problem, the interval [0, 2π) indicates that we are looking for solutions within one full rotation of the unit circle, including 0 but excluding 2π. Understanding how to interpret and apply interval notation is important for determining valid solutions in trigonometric contexts.