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. Start by expressing the equation in terms of a single trigonometric function. Recall the Pythagorean identity: \(\cos^{2} x = 1 - \sin^{2} x\). Substitute this into the equation to rewrite it entirely in terms of \(\sin x\).
2. After substitution, the equation becomes \(10(1 - \sin^{2} x) + 3 \sin x - 9 = 0\). Simplify this expression by distributing and combining like terms.
3. Rewrite the simplified equation as a quadratic in \(\sin x\). It will take the form \(a \sin^{2} x + b \sin x + c = 0\). Identify the coefficients \(a\), \(b\), and \(c\).
4. Solve the quadratic equation for \(\sin x\) using the quadratic formula: \(\sin x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Calculate the discriminant and find the possible values of \(\sin x\).
5. Determine which solutions for \(\sin x\) lie within the valid range \([-1, 1]\). For each valid solution, find the corresponding values of \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the unit circle.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the Pythagorean identity, cos²x = 1 - sin²x, is essential to rewrite the equation in terms of a single trigonometric function, simplifying the solving process.

Solving Quadratic Equations in Trigonometric Form

Many trigonometric equations can be transformed into quadratic form by substituting expressions like sin x or cos x with a variable. Solving the resulting quadratic equation helps find possible values of the trigonometric function, which can then be used to determine the angle solutions.

Interval Restriction and Solution Verification

The problem restricts solutions to the interval [0, 2π), meaning only angles within one full rotation are valid. After solving the equation, it is important to check which solutions fall within this interval and verify them to ensure they satisfy the original equation.