Find the exact value of each expression. sin⁻¹ 1/2

Question

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 expression evaluate and give the answer as an exact value in radiance where the trigonometric expression given is the inverse sign of squared of three divided by two. Now the answer choices provided are a pi divided by two B pi divided by three C pi divided by four and D pi divided by six. Now for a question like this, the first thing that we should do is set up an if and then situation. So if they tell us that the inverse sign of square root of three divided by two is equal to X, let's say then we have the, the regular sign of X is equal to squared of three divided by two. So now that we have this information, let's see how this applies to a coordinate system, see what quadrant this can fall in. So one thing we can know is that sign is positive in our question, this sign is positive and we know that sign is positive in quadrant one and two. And another thing we should note is that the inverse sign is defined inversine being what we are dealing with in this question. So it is defined for a range of negative pi divided by two, two pi divided by two positive pi divided by two. Therefore, this means that we are in quadrants. Well, let's see, we know we're either in quadrate one or two. Now we're only defined between negative pi divided by two and pi positive pi divided by two. Now negative pi divided by two, think of it as the bottom half of the y axis in a coordinate system. And positive pi divided by two is the top half of the Y axis. And this is on located on the right side. So we move counterclockwise here. So if we're moving counterclockwise and we're only defined, that means we're only defined for quadrants on the right side of the Y axis and what quadrants are there? One and four and which one of those two overlaps with our sign being positive quadrants one. So we are in quadrant one. So now that we have that information, let's recall from the unit circle. So now we can use the unit circle to our advantage. So from the unit circle, there are many different angles let's pick, for example, pi divided by three. Now at pi divided by three, there is a specific point that is defined as one half comma squared of three divided by two. Now this is something that we should be able to recall from the unit or a value that we should know. So another thing from the unit circle is that when you have sine of theta, this will be equal to the Y value of that theta. So S sign is associated with the Y values. So if we have, let's say sine of pi divided by three, this will be equal to the Y value at pi divided by three which we just sent will be square root of three divided by two. And now let's compare this to what we had written at the beginning of the question. At the beginning, we had sign of X is equal to a squared of three divided by two. And now we have that sign of pi divided by three is equal to squared of three divided by two. Therefore, the X in our question is equal to pi divided by three because it is in quadrant one and it falls within the range that we are given. So just like that we have answered this question and we know that the answer will be answer choice B so we can see how useful it is to be able be able to know the ranges and use the unit circle to our advantage. So I hope that was helpful. And until next time.

Find the exact value of each expression. sin⁻¹ 1/2

Key Concepts

Inverse Sine Function (sin⁻¹ or arcsin)

Exact Values of Sine for Special Angles

Domain and Range Restrictions of Inverse Trigonometric Functions

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