In Exercises 63–84, use an identity to solve each equation on ...

Question

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin² x - 2 cos x - 2 = 0

Accepted Answer

Hello, today we're going to be solving the following trigonometric equation on the interval from 0 to 2 pi. So what we are given is sine squared of X plus two cosine of X plus two is equal to zero. Now, we'll need to use trigonometric identities in order to simplify the given equation. And we need to simplify sine squared of X. There's a trigonometric identity that states that sine squared of X is equal to one minus cosine squared of X. If we were to use this identity, we can rewrite sine squared of X as one minus cosine squared of X plus two, cosine of X plus two is equal to zero. We can add the constants two and one together. And that will allow us to rewrite the equation as negative cosine squared of X plus two cosine of X plus three is equal to zero. Next, we want to make the leading term positive. And in order to make the leading term positive, we'll need to divide the entire equation by negative one. This well vow was to rewrite the equation as positive cosine squared of X minus two, cosine of X minus three is equal to zero. Now notice that the left hand side of the equation roughly resembles a polynomial. We can use a substitution to rewrite this as a simplified polynomial. So let's say Y is equal to cosine of X. If we were to apply this substitution, we can rewrite the equation as Y squared minus two, Y minus three is equal to zero. The left-hand side of the equation is now in the form of a factor trinomial and this trinomial will factor to give us the quantities Y plus one multiplied by Y minus three is equal to zero. Then using the zero property, we can set both of the factors of Y equal to zero, giving us two equations. We have Y plus one equal to zero and we have Y minus three equal to zero. We need to solve for Y in both of the equations. So we can start by add subtracting one to both sides of the equation. On the left hand side leaving us with Y is equal to negative one. And on the right hand side, we can add three to both sides of the equation leaving us with Y is equal to positive three. If we were to ress substitute cosine back in for Y, we get two equations which states that cosine of X is equal to negative one and cosine of X is equal to positive three. Now, there's one thing to know about the right hand equation. The maximum value that cosine effect can be on the unit circle is positive one but three is greater than positive one. Which means that three lies outside of the region of the unit circle. So since the value of three is outside the unit circle, we're going to disregard the second equation. So what we want to solve is the first equation cosine of X is equal to negative one. Now, in order to answer this, we need to ask ourselves what angles of X can we plug in a cosine that will give us the value of negative one? While in order to answer this, we'll need to use a unit circle. Now recall that any terminal point on the unit circle is defined as cosine comma sine. This is where cosine is any X value of the terminal points and sine is any Y value of the terminal points. So in order to answer this equation, we need to find the angles that have the value of negative one in the exposition of its terminal point. While there is one angle that contains this value in the exposition of the terminal point at the angle pi the terminal point at this angle is negative one comma zero. This terminal point has the value of negative one in the exposition. And what this means is that if we were to plug in the angle pi into cosine, we'll get the value of negative one. So there is only one solution to X for the equation X is equal to pi. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin² x - 2 cos x - 2 = 0

Key Concepts

Pythagorean Identity

Trigonometric Equation Solving

Interval Restriction and Solution Sets

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