Find all solutions of the equation in the interval [0,2 π). 2 sin^2 x=2+cosx

Name: Problem 28, Chapter 5: Precalculus: A Concise Course
Uploaded: 2021-07-06T15:41:22-08:00
Duration: 3 min 13 s
Channel: Monica Miller
Description: Find all solutions of the equation in the interval [0,2 π). 2 sin^2 x=2+cosx

step 1 Okay, we're going to find all the solutions to the given equation on the interval from 0 to 2 pi

Find all solutions of the equation in the interval [0,2 π)....

Key Concepts

Domain Considerations for Trigonometric Equations

When solving trigonometric equations, it is crucial to consider the specified interval—such as [0, 2?)—because trigonometric functions are periodic. This means that one must select the appropriate solutions that fall within the given range, ensuring that all valid answers are identified and none are extraneous.

Solving Quadratic Equations

After transforming a trigonometric equation into a quadratic form, traditional algebraic methods like factoring, completing the square, or using the quadratic formula become applicable. Solving these quadratic equations provides the potential values for the trigonometric function in question.

Pythagorean Identity

The Pythagorean identity, sin²x + cos²x = 1, is a fundamental relationship in trigonometry that allows one to substitute sin²x with 1 - cos²x (or vice versa). This substitution simplifies equations by expressing them in terms of a single trigonometric function, which is particularly useful in solving trigonometric equations.

Reducing Trigonometric Equations to Quadratic Form

Many trigonometric equations can be manipulated into a quadratic form by using identities, such as converting sin²x into 1 - cos²x. Once reduced to a quadratic equation in terms of cos x or sin x, one can apply standard techniques for solving quadratic equations, like factoring or the quadratic formula.

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