Find all solutions on the interval [0,2 π). 2 sin(x) cos(x)-sin(x)+2 cos(x)-1=0

Name: Problem 38, Chapter 7: Precalculus: An Investigation of Functions
Uploaded: 2021-08-11T03:13:06-08:00
Duration: 2 min 40 s
Channel: Luke Waitrovich
Description: Find all solutions on the interval [0,2 π). 2 sin(x) cos(x)-sin(x)+2 cos(x)-1=0

Find all solutions on the interval [0,2 π). 2 sin(x)...

Key Concepts

Solving Basic Trigonometric Equations

After an equation is factored, solving the resulting basic trigonometric equations (such as sin x = -1 or cos x = 1/2) is a key step. This involves knowing the general solutions for sine, cosine, and other trigonometric functions, and then finding those solutions that satisfy the original equation.

Trigonometric Equations

This concept involves setting up and solving equations that include trigonometric functions. It requires identifying how different trigonometric functions interact within an equation, manipulating the equation using known identities, and determining all possible solutions within a given domain.

Trigonometric Identities

Trigonometric identities, such as the double-angle formula (e.g., sin(2x) = 2 sin x cos x), are crucial for transforming and simplifying trigonometric expressions. These identities allow one to rewrite parts of an equation into equivalent forms that are easier to factor or solve.

Factoring Techniques

Factoring is a method of rewriting an expression as the product of its factors. In the context of trigonometric equations, factoring can reveal common terms, simplify the equation, and break it down into simpler sub-equations. This process is essential for isolating trigonometric functions, which can then be solved in a straightforward manner.

Interval Considerations

When solving trigonometric equations, it's important to consider the specified range or domain for the solution. This means only selecting the solutions that fall within the given interval, ensuring that the final answer adheres to the problem's constraints.

Transcript

00:01 Hello and welcome to this video where we're going to solve and find all solutions from 0 to 2 pi for this equation now this is a very tricky equation because we got a lot going on here but what i want to point out to you very quickly is that we have one two three four terms in here if we look at it i see some things that could factor out of some of them but not all of them and if that's the case that means we probably want to start thinking about factored by grouping so take a look at this we're gonna break it up into two halves the yellow half and the green half and so what i've seen in the yellow half is a sign that can be factored out of those two things and so i'll have sign of x times the quantity two cosine of x minus and then one plus nothing really factors out of that one so i'm just going to leave i'm going to do this i'm going to put a one here i'm going to put a parenth or a bracket two cosine of x minus one i just sort of rewrote it.

01:00 I'm multiplying it by one, but it doesn't change its value, but it makes it look like something we should know how to deal with.

01:06 Because notice how, oops, i thought i was on the highlighter.

01:09 There we go.

01:10 That's the highlighter.

01:11 We can take, we can factor that out.

01:14 Those are the same terms.

01:15 So we can pull those out.

01:16 And when we do that, we're going to get, in brackets, 2, cosine of x minus 1 times the quantity of sine of x plus 1.

01:31 And all of this is equal to 0.

01:34 Well, this is an a times b equals 0 type of a situation, so this has to equal 0 or that has to equal 0.

01:42 2, cosine of x minus 1 equals 0, or sine of x plus 1 equals 0...