Find the exact value of each expression. cos⁻¹ (-√2/2)

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 get the answer as an exact value and radiance where the trigonometric expression given is the inverse cosine of negative one half. Now the extras provided are a two pine divided by three B three pi divided by four C five pi divided by four and D pi divided by six. Now for a question like this, let's set up an if and then scenario. So let's say if inverse cosine of negative one half is equal to X, then regular cosine of X is equal to negative one half. Now some things we should note is that cosine regular cosine that is in our question is negative. So we have to think about where in, in, in a coordinate system we are lying at. So cosine is negative in two different quadrants and those will be quadrants two and three. And we can also talk about inverse cosine, which is another part of our question. So we have inverse cosine that is defined or has a range for zero comma pie. So what does this mean it means that we are therefore in quadrants. And let's think about this before we reach what quadrant we are in, if we're defined from zero to pi and we're moving counterclockwise, we are above the X axis. So we are either in quadrant one or quadrant two, we just said that cosine is negative in quadrant two and three. So there's one quadrant there that overlaps in both scenarios and that is quadrant two. So we are in quadrant two. Now, we can recall from the unit circle and the unit circle has a lot of values that we should be able to recall at any moment's notice. So let's recall for example, the a theta of two pi divided by three that will have a point at negative one half comma squared of three divided by two and two pi divided by three lies and quadrant two. Now, something we should know is that cosine of theta is equal to the X values or is associated with the X values. So if we have cosine of two pi divided by three, we look at the X value at two pi divided by three and that is negative one half. And if we said earlier that we're looking for cosine of X equals negative one half, and we just said cosine of two pi divided by three is equal to negative one half. Therefore, X in our question will be equal to two pi divided by three. And just like that, that matches with answer choice A and we can see how important it is to be able to recall different values from the unit circle as well as know the different ranges for our trigonometric functions. So I hope that was helpful.

Find the exact value of each expression. cos⁻¹ (-√2/2)

Key Concepts

Inverse Cosine Function (cos⁻¹)

Exact Values of Cosine for Special Angles

Handling Negative Cosine Values

Watch next