Find the exact value of each expression. cos⁻¹ √3/2

Question

Accepted Answer

Hello everybody. I hope you are 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 that they give us is inverse cosine of square root of two 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, let's set up an if and then situation. So we have that if the inverse cosine of squared of two divided by two is equal to X, then regular cosine of X is equal to the squared of two divided by two. No. When we have this information with a trigonometric function, we can deal with the unit circle and with a cot system. So let's see first in that coordinate system where we can be located. Now, we know that cosine is positive in two quadrants and we're focusing on positive because that's what we have in our problem. We have positive cosine of X. So we are positive and quadrants one and four. Now inverse cosine, which is our actual equation is defined or it has a range between zero and pie. So we need to find, so where this is if we're either in quadrate one or quadrant four, but we're only defined between zero and pi which of those two quadrants is defined between zero and pi Well, we have that therefore, we are in quadrants one. So quadrant one falls within zero to pi quadrant four does not. So with that information, now we can go ahead and use the unit circle to our advantage. So let's recall from the unit circle. Now, in the unit circle, we have different points and the different values corresponding to that point. So in our case, let's say we have the point pi divided by four. Let's recall that at that point, we have a component of squared of two divided by two for the X value comma squared of two divided by two for the Y value as well. So at pi divided by four, we have a point at squared of two divided by two comma squared of two divided by two. Now, from the unit circle, we also know that cosine of theta is equal to the X value. So it's associated with the X value cosine is associated with the X. So if we have, let's say cosine of pi divided by four, we simply have to focus on the X value at pi divided by four, which we see that is squared of two divided by two. Now, this is exactly what we're solving for. We have cosine of X is equal to squared up two divided by two. And we just said that cosine of pi divided by four is equal to squared up two divided by two. Therefore, the in our question is equal to pi divided by four. And just like that we've solved for the answer, which is answer choice C so we can see how important it is to be able to know the ranges and where each function falls in what quadrant to be able to use the unit circle to our advantage. So I hope that was helpful. And until next time.

Find the exact value of each expression. cos⁻¹ √3/2

Key Concepts

Inverse Cosine Function (cos⁻¹ or arccos)

Exact Values of Cosine for Special Angles

Domain and Range Restrictions of Inverse Trigonometric Functions

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