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

Question

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

cos 2x + 5 cos x + 3 = 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 cosine two X plus three cosine of X plus one is equal to zero. In order to solve this equation, we first need to simplify cosine of two X. There's a trigonometric identity that states that cosine of two X is equal to two cosine squared of X minus one. If we use this trivial metric identity to simplify cosine of two X, we can rewrite the left-hand side of the equation as cosine, I'm sorry, two cosine squared of X minus one plus three cosine of X plus one is equal to zero. If we combine the constants negative one and one together, we can eliminate the constants from the equation. And that will leave us with two cosine squared of X plus three cosine of X is equal to zero. Notice that each of the terms have a multiple of cosine. What this means is that we can factor out cosine from the left hand side of the equation and we can rewrite the left hand side as cosine of X multiplied by the quantity two cosine of X plus three is equal to zero. Now, what we can do is we can set the factors of X equal to zero. And that will give us two separate equations cosine of X is equal to zero and two cosine of X plus three is equal to zero. We'll need to simplify the right hand equation. And in order to do that, we can first subtract three to both sides of the equation that will leave us with two cosine of X is equal to negative three. And if we divide both sides of the equation by two, what we are left with is cosine of X is equal to negative three divided by two. Now, one thing to note is that in order to answer this question, we need to ask ourselves what values or what angles of X can we plug in a cosine to give us the value of negative 3/2? Well, one thing to note is that negative 3/2 is equal to negative 1.5. But on the unit circle, the maximum amount that cosine can equal to is one. So what this means is that the value negative 3/2 lies outside of the range of the unit circle. So we're going to disregard this equation and solve for cosine of X is equal to zero. So what we need to ask ourselves is what angles of X can we plug in a cosine to give us the value of zero. Well, in order to answer this, we'll need to use the unit circle. Now recall that any terminal point on the unit circle is defined as cosine comma S sign. This is where cosine is the X value of the terminal point and sine is the Y value of the terminal point. So in order to answer cosine of X is equal to zero, we need to find the angles on the unit circle that have zero in the X value of the terminal point. While there are two angles that have zero in the X value of their terminal points, those angles are going to be a pi over two. And at three pi over two, the terminal points at pi over two is zero comma one and the terminal point at three pi over two is zero comma negative one. What this means is that able to plug in both of these angles into cosine will get the values of zero. And that means that there are two solutions for XX is equal to pi over two and X is equal to three pi over two. And with that being said, the answer to this problem is going to be C 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). cos 2x + 5 cos x + 3 = 0

Key Concepts

Trigonometric Identities

Quadratic Equations

Interval Notation

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