Solve (1-sin^2 x-2 cosx)/(cos^2 x-cosx-2)=-(1)/(3) algebraically over the domain -π≤x ≤π..

Name: Problem 18, Chapter 6: Precalculus
Uploaded: 2021-08-19T01:14:42-08:00
Duration: 5 min 43 s
Channel: Matthew Markham
Description: Solve (1-sin^2 x-2 cosx)/(cos^2 x-cosx-2)=-(1)/(3) algebraically over the domain -π≤x ≤π..

Solve (1-sin^2 x-2 cosx)/(cos^2 x-cosx-2)=-(1)/(3)...

Key Concepts

Solving Trigonometric Equations

Once the equation has been simplified, solving a trigonometric equation such as cos x = a constant requires knowledge of how to determine all solutions within a specified interval. This involves finding the principal value and leveraging the periodic nature of trigonometric functions to capture all possible solutions within the given domain.

Domain and Range Restrictions

When manipulating expressions and canceling factors, it is critical to consider the inherent restrictions on trigonometric functions. In this context, understanding that cosine values are constrained to the interval [-1, 1] helps avoid introducing extraneous solutions, ensuring that any canceled factors do not correspond to valid solutions that are lost.

Trigonometric Identities

This concept involves using standard relationships between sine and cosine functions, such as the Pythagorean identity sin²x + cos²x = 1. Such identities allow for the substitution and simplification of trigonometric expressions, reducing the equation to a more manageable form.

Algebraic Manipulation and Factorization

This key concept focuses on the ability to manipulate algebraic expressions—including factoring common terms—in order to simplify equations. By factoring the numerator and the denominator and canceling common factors (while being mindful of domain restrictions), one can reduce the equation to a simpler form that is easier to solve.

Transcript

00:01 All right, so we have this equation and we want to solve it algebraically.

00:09 So the first step we're going to do is we're going to take everything on the bottom side times the entire thing by it.

00:16 So it cancels it on the left -hand side and moves it to the right -hand side.

00:19 And they do the exact same thing with the left -hand side.

00:22 So since this on the top is just going to be one and then it's just going to be kos.

00:29 Just everything here is just going to go on the right -hand side and everything here is going to go.

00:32 Going to be times by minus 3.

00:34 So that's going to give us minus 3 plus 3 sine squared x plus 6 cos x equals co squared x minus cos x minus 2.

01:09 So we want to, so we can move the, we can move the minus 3.

01:24 Three here to this side and the six cos x to that side.

01:34 And so that's gonna give us co squared squared x minus seven cosine of x.

01:48 Minus two is still over here.

01:51 And then it's going to be plus one there.

01:59 Now for this side, it's still gonna be three sine squared of x, but we all know, but since we're going to be putting everything in cosine, we're going to take sine squared is going to be equal to 1 minus sine squared, just from the pythagorean identity right here.

02:19 So it's going to give us 3 minus 3 cosine squared of x...

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