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).4 cos² x = 5 - 4 sin x

Accepted Answer

Hello, today we're going to be solving the following trigonometric equation from the interval of 0 to 2 pi. So what we are given is two cosine squared of X is equal to two minus three of sine of X. Now, in order to simplify this equation, we'll need to use a trigonometric identity to simplify cosine squared of X. We have a trigonometric identity that states that cosine squared of X is equal to one minus sine squared of X. So if we use this trigonometric identity, we can rewrite the quantity cosine squared of X as one minus sine squared of X. And that will allow us to rewrite the equation as two multiplied by the quantity one minus sine squared of X is equal to two minus three sine of X. In order to further simplify the quantity, we're going to distribute two into one minus sine squared of X. And that will leave us with two minus two sine squared of X is equal to two minus three sine of X. We can subtract two to both sides of the equation to eliminate the constants that will leave us with negative two sine squared of X. Is equal to negative three sine of X. Next, we want to go ahead and set the equation equal to zero in order to solve for X. So we're going to add three sine X to both sides of the equation. Doing so allows us to rewrite the equation as negative two sine squared of X plus three sine of X is equal to zero. Now, one thing to note is that both of the terms on the left hand side of the equation have a multiple of sine. So we can factor out sine from the left hand side of the equation and rewrite the left hand side as sine of X multiplied by the quantity negative two sine of X plus three is equal to zero. We can use the zero property to create two equations and that will leave us with sine of X equal to zero and negative two multiplied by sin of X plus three is equal to zero. We'll need to simplify the right hand equation. In order to simplify the right-hand equation, we'll subtract three to both sides leaving us with negative two, multiplied by sin of X is equal to negative three. We'll then divide both sides of the equation by negative two. And that will leave us with sign of X is equal to positive three divided by two. Now, the right-hand equation is saying that sin of X is equal to positive three divided by two. But one thing to note is that the maximum value for sin of X on the unit circle is one, since 3/2 is greater than one, we're going to ignore this right-hand equation since the value 3/2 does not lie within one region or lies outside of the unit circle. So what we are left with is sign of X is equal to zero. And now we need to ask ourselves what angles of X can't. We plug in a sign that will give us the value of zero. Well, in order to answer this problem, we'll need to use a unit circle. Now recall that any terminal point on the unit circle is defined as cosine comma S sign. This is where cosine is any X value of the terminal points and sine is any Y value of the terminal points. So what we need to ask ourselves is what angles in the unit circle have zero in the Y position of the terminal point? Well, there are two angles that give us the value of zero in the terminal point. The first angle is zero and the terminal point at this angle is one comma zero. The second angle will be pi and the terminal point at pi is negative one comma zero. Notice that both of the terminal points have the value of zero in the Y position. What this means is that if we were to plug in the angles of zero and pi into sine we'll get the value of zero. That means that there are two solutions to X for the equation X is equal to zero and X is equal to pi. And with that being said, the answer to this problem is going to be D. 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).4 cos² x = 5 - 4 sin x

Key Concepts

Trigonometric Identities

Pythagorean Identity

Solving Trigonometric Equations

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