Evaluate cos (11π/12) as cos (π/4 + 2π/3).

Question

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the value of cosine of 5 pi divided by 12 as cosine of pi divided by 6 plus pi divided by 4. Awesome. So it appears for this particular problem we're ultimately trying to figure out the value of cosine of 5 pi divided by 12 as cosine of pi divided by 6 plus pi divided by 4. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be, noting that they all state that cosine of 5 pi divided by 12 is equal to some expression. So A is the square root of 6 plus the square rooted 2, all divided by 2. B is the square root of 6 minus the square rooted 2, all divided by 2. C is the square root of 6 plus the square rooted 2 divided by 4. And finally, D is the square root of 6 minus the square rooted 2, all divided by 4. Awesome. So first off, in order to find the value of cosine of 5 pi divided by 12 in terms of cosine of pi divided by 6 plus pi divided by 4, we need to recall and use the cosine addition formula. So as we should recall, the cosine addition formula states that cosine of A plus B. is equal to cosine of a multiplied by. Oh sign of B. Minus sign of a multiplied by sign of B. Fantastic. So now let us define our variables. Let us state that A is going to be equal to pi divided by 6, and let us state that B is going to be equal to pi divided by 4. So now, if we plug these values back into our expression for the cosine addition formula, we will find that cosine of pi divided by 6. Plus, I divided by 4 is going to be equal to Drum roll, cosine of pi divided by 6 multiplied by cosine of pi divided by. 4 Minus sign of pi divided by 6 multiplied by sign of pi divided by 4. Fantastic. So now, our next step is we need to recall and use the trigontric functions to help us solve for the different values for cosine and sine. So, as we should recall, let us note that cosine of pi divided by 6 is going to be equal to the square root of 3 divided by 2. And let us also note that cosine of pi divided by 4 is going to be equal to the square rooted 2 divided by 2. Let us also note that sine of pi divided by 6 is going to be equal to 12 and sign of pi divided by 4 is going, so once again, sign of pi divided by 4. is going to be equal to, as we should recall, square rooted 2 divided by 2. Fantastic. So now we need to substitute these values back into our formula. So when we do that, we will find that cosine of pi divided by 6 plus pi divided by 4 is going to be equal to the square root of 3 divided by 2, multiplied by the square root of 2, divided by 2. Minus 12 multiplied by the square root of 2 divided by 2. So, when we simplify this expression, we will find that the cosine of pi divided by 6 plus pi divided by 4. Fantastic. So, once again, the cosine of pi divided by 6 plus pi divided by 4 is going to be equal to the square root of 6 divided by 4 minus the square root of 2 divided by 4. Which we can also write as it's equal to the square root of 6 minus the square root of 2, all divided by 4. And that's it. That is our final answer. Hooray, we did it. So, therefore, without a doubt, the value of cosine of 5 pi divided by 12 is going to be the square to 6 minus the square to 2 all divided by 4. And this corresponds with multiple choice answer letter D, which once again is the cosine of 5 pi divided by 12 is equal to the square rooted 6 minus the square rooted 2, all divided by 4. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Evaluate cos (11π/12) as cos (π/4 + 2π/3).

Key Concepts

Cosine Function

Angle Addition Formula

Unit Circle

Watch next