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 look 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 negative square root of two divided by two. Now the answer choices provided are a negative pi divided by two B negative pi divided by three C negative pi divided by four and D negative pi divided by six. Now for a question like this, let's set up an if and then situation. So let's say if inverse sign of negative square root of two divided by two is equal to X, let's say then regular sign of X is equal to negative square root of two divided by two. So we're trying to solve for that X. Now two things we should note is that sign in our case, the sign is positive. So sign is positive or negative. I'm sorry. The answer for our sine sine of X is negative squared of two divided by two. So Sine is negative S is negative in two quadrants and those are quadrants three and four So we know we are either in quadrant three or quad four. Now for inverse sign, we have the inverse sign is defined for a range from negative squared or negative pi divided by two comma pie, positive pi divided by two. Therefore, we are in quadrant. And let's think about this for a second. Well, if I range negative pi divided by two, that is the bottom half of the Y axis and that goes to positive pi divided by two, which is the top half of the Y axis. And we're moving in a counterclockwise rotation. So if we start at the bottom half of the Y axis and end at the top half of the Y axis, and we're moving counterclockwise, we pass through quadrants four and quadrant one. Now which one of those quadrants matches up with the ones we had previously mentioned that is quadrant four. So we are located in quadrant four. Now that we have this information, we can use it to recall from the unit circle and use the unit circle to our advantage. So if we recall from the unit circle, let's say the a theta of pi divided by four. So these are values that we should be able to recall. Now, at an angle of pi divided by four, we have a point of square root of two divided by two comma square root of two divided by two. Now something we should note is that sign of theta is equal to or is associated with Y values. So if we have, let's say sine of pi divided by four, this will be equal to the Y value associated with a theta of pi divided by four, which we just said was squared of two divided by two. Now we had that sign of X is equal to negative squared of two divided by two. So we need the negative value of what we just calculated. So let's say if we multiply both sides by a negative, we'll have negative sign pi divided by four is equal to negative squared of two divided by two. Now, we should know that negative sign of theta is equal to a sign of negative theta. Therefore, we have that negative sign of pine divided by four is equal to a sign of negative pi divided by four. And this will be equal to negative square root of two divided by two. Therefore, the in our question is negative pi divided by four because negative pi divided by four falls in quadrant four and the range that we are looking for. So just like that we solved for the X which is and matches up with answer choice C and we can see how useful it is to be able to use the unit circle to our advantage as well as knowing the different ranges of our trigonometric functions. 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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