Find the exact value of each expression. (Do not use a calculator.)

cos(-15°)

Video transcript

Welcome back. Everyone. In this problem, we want to determine the exact value of the cosine of negative 75 degrees without using a calculator. For our answer choices. A says is the square root of six minus the square root of two all divided by four B says is the square root of six plus the square root of two all divided by four C is the square root of six minus the square root of two all divided by two and B is the square root of two minus the square root of six all divided by four. Now first, let's talk about our trigonometric function a little bit. Now here notice we're finding the cosine of a negative angle but recall, OK, that the cosine of a negative angle is equivalent to the cosine of the positive value of that anger. Therefore, this tells us then that the cosine of negative 75 degrees is the same thing as the cosine of 75 degrees. OK? No, we don't know the exact value of the cosine of 75 degrees, but we know that 75 degrees is made up of values that we know their exact trigonometric functions. What I'm really saying here is that we could rewrite the cosine of 75 degrees as the cosine of 30 degrees plus 45 degrees. So we know that the sum of 75 is 30 degrees and 45 degrees. And we know those exact values. Now, what do we know about the sum of our, of our angles for trigonometric function? Well, we also know that by the some identity, the core sign of the sum of two angles is going to be equal to the cosine of one angle multiplied by the cosine of the other angle minus the sine of one angle multiplied by the sine of the other angle. OK. So what does that tell us then? Well, no, that means for our trigonometric function that we have here, we can use the sum identity to rewrite it and then figure out the exact values for the cosine and sine functions for those angles. So that means then that the cosine of 30 degrees plus 45 degrees, it is going to be equal to the cosine of 30 degrees multiplied by the cosine of 45 degrees minus the sine of 30 degrees multiplied by the sine of 45 degrees. No, this problem is easier to work with because we can find the exact values of all of these functions. The cosine of 30 the cosine of 45 the sine of 30 the sine of 45. Now to help us, let's use our special triangles for a 30 degree angle and a 45 degree angle. Now recall, let's put that here that for a 45 degree angle, any, any triangle with a 45 degree angle can basically be seen as a similar triangle to this special triangle. A 45 degree triangle unit triangle where both of its sides are one unit and its hypotenuse is the square root of two. Likewise, we also know that for our 30 degree angle, OK. For a 30 degree angle, it also forms a special triangle or a special right triangle as well. Because for this triangle, its hypotenuse is going to be two units. The side opposite to our 30 degree angle is one unit and opposite to a 60 degree angle is going to be equal to the square root of three. So we can use both of these to find the exact values for our cosine and sine functions because recall as well that the sine function is the ratio of the opposite side to the high party news. And the cosine function is equal to the or is the ratio of the adjacent side to the hypotenuse. So no, oops, I don't know. Let me put that fix that properly here. So no, what that tells us then is that for our function for our sum, the cosine of 30 degrees plus 45 degrees, what we're really saying here is that to find a cosine of 30. When we go to our 30 degree triangle, the cosine of 30 is going to be equal to the side that's adjacent to 30 degrees, which is the square root of three, two, the hypotenuse which is two. That's the cosine of 30. Let's do that for the rest of Fengs here. OK. Now, if you want to find a cosine of 45 degrees, let's go to our, let me write that here. Let's go to our 45 degree triangle. And the cosine is going to be equal to the adjacent side to the 45 degree angle which is one divided by the hypotenuse, which is the square root of two. Likewise for the sine of 30 degrees, it is going to be equal to its opposite side which is one divided by the hypotenuse which is two. And then finally, the sign of 45 degrees is going to be equal to its opposite side, which is one. So here I think I call this adjacent earlier. So this the adjacent would actually be here that's adjacent to the 45 degree angle, my apologies, but the opposite side to the 45 degree angle is also one. OK. And its hypotenuse is the square root of two. So now we have the exact values for all of those trigonometric functions. All we need to do now is to simplify and show which of our answers it would have been equal to. So let's go ahead and do that. Now here, we can rationalize our expression for this one divided by the square of two. OK? Because when we multiply both terms, let me put that over in a corner here because we could multiply both multiplied by the square of two, divided by the square of two and one multiplied by the square of two equals the square of two. While the square of two multiplied by the square root of two simply equals two. So we can replace that in our expression to know how the skirt of three multiplied. Sorry, the skirt of three divided by two multiplied by the skirt of two divided by two minus a half multiplied by the square of two divided by two. Now, if we simplify the spirit of three multiplied by the square of two equals the square of six and two multiplied by two equals 41 multiplied by the square of two equals the square of two. While two multiplied by two equals four, both of them have the same denominator. So we can rewrite it as the square of six minus the square root of two divided or all divided by four. Therefore, the cosine of 75 degrees or the cosine of negative 75 degrees would be equal to the square of six minus the square of two, all divided by four. If we look back on our answer choices, that means a would be the correct answer. Thanks a lot for watching everyone. I hope this video helped.