Evaluate cos⁻¹(cos(5π/4)).

Question

Accepted Answer

Welcome back everyone. In this problem, we want to find the value of the inverse cosine of the cosine of 19 17th of pi. For our answer choices, a is 19 17th Softpie, b is 15 seventeenth Softpie. C is 2 seventeenth Softpie. And the d is negative 19 seventeenth of pie. Now, what are we trying to figure out here? Well, basically what we're doing is putting the function cosine into its inverse function. And recall that for functions, if we put a function into its inverse, it gives us our input value, the value we started with. And that makes sense because if you really think about it, let's say that I have a value x that I'm putting into a function. So, I apply f and then I get f of x. Likewise, in the same way, if I were to undo that then I'd apply the inverse function f inverse of x. That would give me where what I started with. And, the cosine function works in the same way. We take an angle x, apply the cosine from which we'll get a ratio, the cosine of x. So if I apply the inverse, essentially, what I'm doing is that I'm taking that ratio, the cosine of x, and then I'm finding out what angle I started with. So right away, it would appear that the solution to our problem is 19 seventeenth of Pi. But the problem with that is that the cosine of an angle has to be within the range from 0 to Pi. So, what that tells us then is that since 19 17th of pi is greater than pi, we need to find a reference angle that would be less than pi. Now, how can we do that? Well, there's another property of the cosine we can also use. Because recall again that the cosine of an angle x equals the cosine of 2 PI minus x. So if we can subtract our value 19 17ths of PI from 2 PI, then we should be able to get a value that is less than Pi. Since 19 17ths of Pi is greater than Pi. So, let's go ahead and do that. So, this tells us then that the cosine of 19 seventeenth of Pi would be equal to the cosine of 2 Pi minus 19 seventeenth of Pi. If we use that idea, 2 Pi minus 19 seventeenth of Pi. Let's go ahead and find a common denominator which is 17. We write 2 Pi as 2 Pi divided by 1. 1 into 17 goes 17 times. So it multiply 2 pi by 17, which would have been 34 pi. Next, 17 17 into 17 goes one time. 19 pi multiplied by 1 equals 19 pi. So that's 34pi minus 19pi divided by 17pi. And 34pi minus 19pi equals 15pi. So our result would be the cosine of 15 seventeenth of pi. Now we have an angle. Is this less than pi? Well, yes. 15 seventeenth of pi is less than pi. So, that means we can then rewrite our problem which was the inverse cosine of the cosine of 19 17ths of pi as the inverse cosine of the cosine of 15 seventeenth of Pi. And now, using that idea, as we said, if we find the inverse function of the same function, it gives us the value we started with which would have been 15 seventeenth of Pi. That would have been our final answer And when we look back in our answer choices, that would have been answer choice b. Thanks a lot for watching everyone. I hope this video helped.

Evaluate cos⁻¹(cos(5π/4)).

Key Concepts

Inverse Trigonometric Functions

Cosine Function and Its Periodicity

Quadrants and Reference Angles

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