In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 2 cos² x + sin x - 1 = 0

Verified step by step guidance

1. Start by recognizing that the equation contains both \( \cos^2 x \) and \( \sin x \). Use the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \) to rewrite the equation entirely in terms of \( \sin x \).
2. Substitute \( \cos^2 x \) with \( 1 - \sin^2 x \) in the equation: \( 2(1 - \sin^2 x) + \sin x - 1 = 0 \).
3. Simplify the equation by distributing and combining like terms: \( 2 - 2\sin^2 x + \sin x - 1 = 0 \) which simplifies to \( -2\sin^2 x + \sin x + 1 = 0 \).
4. Multiply the entire equation by \( -1 \) to make the quadratic term positive: \( 2\sin^2 x - \sin x - 1 = 0 \). Now, treat \( \sin x \) as a variable (say \( t \)) and solve the quadratic equation \( 2t^2 - t - 1 = 0 \).
5. Find the roots of the quadratic equation using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a=2 \), \( b=-1 \), and \( c=-1 \). Then, for each root \( t \), solve \( \sin x = t \) on the interval \( [0, 2\pi) \).

Key Concepts

Pythagorean Identity

The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This fundamental relationship allows us to express cos²x in terms of sin²x or vice versa, which is useful for rewriting trigonometric equations into a single function to simplify solving.

Solving Quadratic Trigonometric Equations

Trigonometric equations involving squared terms can often be treated like quadratic equations by substituting a trigonometric function with a variable. This approach helps factor or use the quadratic formula to find solutions within the given interval.

Interval Restriction and Solution Verification

When solving trigonometric equations, solutions must be found within the specified interval, here [0, 2π). After solving, it is important to verify that all solutions fall within this range and to consider the periodic nature of trigonometric functions.