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

Question

In Exercises 11–24, find all solutions of each equation.                   1cos x = ﹣-------                   2

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 the trigonometric equation is cosine of D is equal to negative open parentheses squared of three closed parentheses divided by two. Now the answer choice is provided are ad is equal to pi divided by three plus two. Pi N or D is equal to four pi divided by three plus two pi N where N is an integer and it N will be an integer for all the answer choices provided as a choice B says that D is equal to two pi divided by three plus pi N or D equals four pi divided by three plus pi N C says that D is equal to five pi divided by six plus two. Pi N or D is equal to seven pi divided by six plus two pi N. And as your choice, D says that D is equal to pi divided by six plus pi N or D is equal to seven pi divided by six plus pi N. Now, for this question, they told us of a trigonometric equation and we're trying to solve for the specific angle. Now, the first thing I'm going to do here after I rewrite our equation. The first thing we can do is simple recall. So recall values from the unit circle that we should know. We can recall that cosine of five pie divided by six is equal to negative square root of three divided by two, which is the answer we're looking for. So we know that therefore five pi divided by six is a solution. So now we want to see if this is the only solution or if there are other ones. Now to do that, let's first think of the possible quadrants that can have a solution quadrants in a graph. Now, the cosine that the angle that we have here, we know that the cosine of angle D is negative. So where is cosine negative? Well, cosine is negative in two different quadrants. Now, when you think of cosine, think of X values in a graph. So where are X values negative? Well, they're negative to the left of the Y axis. Therefore, cosine is negative in quadrants two and three because that's where X values are negative. Now, we also want to look for the reference angle of the angle that we resolved for or of our current solution. So we want to solve for the reference angle of five pi divided by six. Now five pi divided by six is less than six pi divided by six whi which is pi, which is, which means that five pi divided by six falls in quadrant two. So we're solving for a reference angle of an angle in quadrant two. And for that, we have a rule that the reference angle will be equal to pi minus the given angle which in this case is five pi divided by six. So we'll have six pi divided by six minus five pi divided by six, which means that our reference angle is pi divided by six. And I'm going to move my work a bit. So I have a little more space and I can go ahead and move on to the other solutions. Now, we already have a solution in quadrant two. So we now might have a solution in quadrant three. So for quadrant three, we have another reference angle rule. So in quadrants three, we have that the reference angle is equal to the given angle or the angle we're trying to solve for here minus pi. Now remember we already solved for the reference angle. So we have that our reference angle pi divided by six is equal to the angle that we want minus pi, which means that we now have pie divided by six plus pi is equal to the angle we're looking for. Now pi can be rewritten as six pi divided by six. So we'll have seven pi divided by six is equal to the angle we are looking for. So our second solution now when we look at a graph of cosine, we know that it can repeat. So they usually have kind of a graph looking like this kind of like a wave that never ends. So those values will repeat as you go under the X axis above under above and so on. So the values are going to repeat as you go forward. So we have something that is called a period, which is how often does that, that generous shape of the graph repeat. So in the period of cosine, we have that, that period is tupai. So every two pi units, the graph of cosine repeats. So now we have two solutions for our angle D, we have, we said already that D is equal to five pi divided by six and D is also equal to seven pi divided by six. Now we have to take into account the fact that the graph continues to repeat forever. So we have to add the period. So we have plus two pi infinitely. So plus two pi N and we add that to both answers. And that would just represent the fact that the graph will repeat and will have other answers infinitely as you move forward with the graph. And therefore these will be your two answers. D equals five pi divided by six plus two pi N or D equals seven pi divided by six plus two. Pi N and this matches with answer choice C for this question. And we can see how important it can be to know how to recall unit circle values as well as understand how to calculate the different signs of each trigonometric function as well as use different rules like the reference angle formulas and so on. So I hope that was helpful. And until next time.

In Exercises 11–24, find all solutions of each equation. 1cos x = ﹣------- 2

Key Concepts

Cosine Function

Inverse Trigonometric Functions

General Solutions of Trigonometric Equations

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