Find the exact value of each real number y. Do not use a calcu...

Question

Find the exact value of each real number y. Do not use a calculator.

y = cos⁻¹ (―√2/2)

Accepted Answer

Hello. Today we are going to be determining the exact value of the given inverse trigonometric function without using a calculator. So we are given Y is equal to cosine inverse of negative one divided by two plus cosine inverse of negative one. Before we jump into this problem, we need to understand the domain of the cosine inverse function. The domain of the cosine inverse function is given to us as negative one comma one. What this means is that on the unit circle, the domain of cosine inverse is going to be the upper half of the unit circle. Now now that we understand where the domain of cosine inverse lies, let's go ahead and evaluate each of the cosine inverse functions starting with cosine inverse of negative one divided by two. Now recall that every time we evaluate an inverse trigonometric function, the output will always be some angle theta. So what we can do is we can take cosine inverse of negative one divided by two and set it equal to theta. If we take the cosine function of both sides of this equation, we can rewrite the equation to be negative one divided by two is equal to cosine of theta. And if we reorder the terms of the equation, we can rewrite the equation as cosine of the is equal to negative one divided by two. Now we need to ask ourselves, what is an angle theta that I can plug into the cosine function that will give me the output negative one divided by two and is also within the domain from negative 1 to 1. Well at the angle two pi divided by three cosine is equal to negative half. So what this means is that cosine inverse of negative one divided by two is equal to two pi divided by three. This is going to be the value of the first inverse trigonometric function. Now we are going to repeat this process with cosine inverse of negative one. We will take cosine inverse of negative one and set it equal to the, if we take the cosine function of both sides of this equation, we can rewrite the equation to be negative one is equal to cosine of the. And if we reorder the terms, we can rewrite this equation to be cosine of the is equal to negative one. Now again, we want to ask ourselves what is an angle theta that I can plug into cosine that will give me the output of negative one. Well, at the angle pi cosine is equal to negative one. So what this means is that the inverse function cosine inverse of negative one will equal to pi this is the value of the second inverse trigonometric function. Now that we solved for both of the values of cosine inverse we can now solve for the main equation. So again, we are asked to solve for Y is equal to cosine inverse of negative one divided by two plus cosine inverse of negative one. We can rewrite this equation to be wise equal to two pi divided by three plus pi. In order to add these two values together, we must have the same denominator. We can represent pi as a fraction by dividing it by one. And we can also multiply this value by three divided by three. This will allow us to rewrite the equation as two pi divided by three plus three pi divided by three and adding the two fractions together will give us five pi divided by three, which is going to be the exact value of the equation without using a calculator. And with that being said, the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Find the exact value of each real number y. Do not use a calculator.
y = cos⁻¹ (―√2/2)

Key Concepts

Inverse Trigonometric Functions

Unit Circle

Reference Angles

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