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.
              __
            √ 3
sin x = -------
              2

Accepted Answer

Hello everybody. I hope you are 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 trigonometric expression is sign of A is equal to one half. Now, the answer choices provided are A, A is equal to pi divided by six plus two pi N or A is equal to five pi divided by six plus two pi N where N is an integer and N will be an integer. And all the answer choices I will provide, B says that A is equal to pi divided by three plus two pi N or A is equal to two pi divided by three plus two. Pi N answer choice C says that A is equal to pi divided by six plus two pi N. And answer choice D says that A is equal to pi divided by three plus two pi N. Now, the first thing we can do right away is use simple recall from our unit circle values. And we can recall that sign of pi divided by six is equal to one half, which means that pi divided by six is a solution. And just like that, we have our first solution by doing simple recall of the unit circle values. Now to solve for other solutions, we have to do a little bit of more thinking. So first, let's look at the sine equation, we have that sign of A is equal to one half, which is a positive one half. So we have to think about where is sine positive. Well, Sine is positive in two different quadrants. Now, when you think of sign, think of why values? So think of Y values in a graph where are the Y values positive? Well, if it's above the X axis, the Y is positive and what two quadrants are above the X axis, those are quadrants one and two. So we have that sinus positive in quadrants one where we already have a solution pi divided by six lies in quarter one and quadrant two where we don't have a solution yet. So we are focusing on a solution for quadrant two. Now I'm going to move my work a bit. So I have a little more room to work with and now let's focus on quadrant two. So in quadrant two, we have a rule for the reference angle. So we have that the reference and winner abbreviated as R E F, we have that reference angle is equal to pi minus the angle that we're looking for. For the reference angle we already obtained that pi divided by six pi divided by six is a reference angle. So we have that pi divided by six is equal to six pi divided by six. And that's just a way of rewriting pi to have the same numerator as our pi divided by six fraction. So we'll have pi divided by six equals six pi divided by six minus the angle we're trying to solve for. So we have that our angle is equal to six pi divided by six minus pi divided by six, which means that we have our angle is equal to five pi divided by six. And this will be our second solution. Now, when we're thinking about the graphs of trigonometric functions, we have to take into account that tho those graphs can repeat and they can go on forever. Now, now the period of a trigonometric function in this case, the period of sign determines how often that graph will repeat. Now the period of sign is to pie. So every two pi units that graph will repeat and will have more answers. So we have to take that into account to our answers. So we had two answers. We had that A is equal to pi divided by six and that A is equal to five pi divided by six. Now to add the other periods as we move forward in a graph, the infinite periods, we just add the two pi period multiplied by an integer. And we add that to both answers. And by doing that, we can take into account all possible solutions for this question. And just like that we've solved for the solutions. And these two solutions match up with answer choice. A and we can see how important it is to be able to recall from the unit circle as well as know the different rules for the reference angle and the signs of the different functions. So it's very important to know how to work with the unit circle. So I hope that was helpful. And until next time.

In Exercises 11–24, find all solutions of each equation. __ √ 3 sin x = ------- 2

Key Concepts

Trigonometric Functions

Inverse Trigonometric Functions

Periodic Nature of Trigonometric Functions

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