In Exercises 11–24, find all solutions of each equation.
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Question

In Exercises 11–24, find all solutions of each equation.
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2 cos x + √ 3 = 0

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today we're going to be looking at this question that states for the following trigonometric equation solve and give the answer in radiance where our equation is four multiplied by cosine of B plus the square root of eight is equal to zero. Now, the answer choices provided are A B is equal to three pi divided by four plus two. Pi N or B is equal to seven pi divided by four plus two pi N or N is an integer and N is an integer for each of the answer choices that I will read. So we have answer choice B that says B is equal to pi divided by four plus two. Pi N or B is equal to five pi divided by four plus two by N C says that B is equal to pi divided by four plus two pi N or B is equal to three pi divided by four plus two pi N. And finally, as choice D says that B is equal to pi or three pi divided by four plus two. Pi N or B is equal to five pi divided by four plus two pi N. Now the first thing we're going to try to do with our equation is simplify it a bit to have cosine by itself. So the first thing we'll do is move the squared of eight from the left hand side of our equation to the right hand side. So we have four cosine of B is equal to negative square of eight. Now, the negative squared of eight can be simplified and we'll have a four cosine being or four multiplied by the cosine of B is equal to negative square root of. And we'll change the eight as four multiplied by two because we know that four has a perfect square root. And we can simplify that as four multiplied by cosine of B is equal to negative two multiplied by the squared of two because the four has a squared of two. So we take that two out and we're left with a two inside of the square root. Now, we divide both sides by four to isolate cosine of B. And we'll have that cosine of B is equal to negative two multiplied by the square of two in the numerator. And the denominator will have four, which means that cosine of being is equal to negative square root of two divided by two. Now, I'm going to move my work a bit to the side to have a little bit of more room to work with. And right away, we can use recall simple recall from our unit circle values. And we can recall that cosine of three pi divided by four is equal to negative squared of two divided by two, which means that therefore three pi divided by four is a solution. Now, for other solutions, we're going to do a little bit of more work. First, let's see where we are what quadrant we are in, in our, in a graph. So let's look at our cosine equation and we see that are the cosine of an angle is equal to negative squared of two divided by two. So we know our cosine is negative. So cosine is negative in two different quadrants. And when we're talking about cosine, we're thinking about X values. So where are X values negative? Well, we know that they are negative to the left of the Y axis. And what quadrant are those? Well, those are quadrants two and three. So cosine is negative in quadrant two and three. Now we have three pi divided by four as one of our solutions which lies in quadrants two. Therefore, for our other solution, we'll focus on quadrant three. Now let's try to solve for the reference angle of pi of three pi divided by four, which was our first solution. Now recall, let's remember that three pi divided by four, we set lies in quadrants two. So for a reference angle of an angle that lies in quadrant two, we use the formula I minus angle. So we have that we have pi minus three pi divided by four. That was the angle that we were given. So we'll have four pi divided by four, which is pi minus three pi divided by four. So we have that our reference angle will be pi divided by four. And I'll move my work once again and we can move on to solve for other solutions. Now that we have a reference angle, we can go ahead and use that to our advantage. Now for quadrant three, so we solved for the reference angle of three pi divided by four, which was in quadrant two. Now we're trying to solve for an angle in quadrant three. So for quadrant three, we have that reference angle is equal to angle minus pi. Now remember that we already solved for our reference angle in this case. So our reference angle is pi divided by four. So we have pi divided by four is equal to the angle we're trying to solve minus pi, which means that we have now pi divided by four plus pi is equal to the angle we're trying to solve for which means that five pie divided by four is equal to the angle and will therefore be a solution. So now before we go to our final answer, let's remember the period of cosine. And why do we not want to know that was as you can see in our answers, we have, for example, two pi N as with the integer added to an angle. Now the period of cosine is two pie. Now this means that every two pie, the graph of cosine will cut will repeat. So how does that affect our answers? Well, we have two solutions. We have B equals something and B equals another thing. So for the first answer, we calculated that three pi divided by four was a solution. So we have B equals three pi divided by four. But if we look at a graph of cosine, we know that it repeats every two pi so every two pi and there will be another solution at the same three pi divided by four. So we have three pi divided by four plus two pi N to represent that those infinite solutions that go on. And we do the same thing for our second solution, which is B equals five pi divided by four. And we'll also have plus two pi N for the same reasons. So that will be the answer or those two will be the answers to this question which we can see matches up with answer choice D. So we can see how important it is to be able to recall different values on command from our unit circle as well as understanding how the different trigonometric functions work in that environment, how, where they are negative, where they are positive and how to get their reference angles. So I hope that was helpful. And until next time.

In Exercises 11–24, find all solutions of each equation. __ 2 cos x + √ 3 = 0

Key Concepts

Cosine Function

Solving Trigonometric Equations

Quadrants and Angle Solutions

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