Find the exact value of the expression.cos⁡5π12\cos\frac{5\pi}{12}cos125π

6 − 2 4 \frac{\sqrt6-\sqrt2}{4} 4 6 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> − 2 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Find the exact value of the expression. cos ⁡ 5 π 12 \cos\frac{5\pi}{12} cos 12 5 π

In this problem, we're asked to find the exact value of the expression, the cosine of 5 pi over 12. Now seeing as I don't know this value off the top of my head, but this is a multiple of pi over 12, I can go ahead and break this down into angles that I do know and then use my sum and difference formulas. So here I want to break down 5 pi over 12 into 2 angles that I do know. Now I know that 5 pi over 12 can be broken up into 2 pi over 12 plus 3 pi over 12. Now simplifying these, this simplifies to pi over 6 and pi over 4, which are two angles that I do know. So here I can find the cosine of pi over 6 plus pi over 4 using my sum formula for cosine. So let's go ahead and do that here. Now expanding this out, I have the cosine of that first angle pi over 6 times the cosine of that second angle pi over 4. Then I'm subtracting of that first angle pi over 6 times the sine of that second angle pi over 4. Now we know all of these trig values, right, from our unit circle. So the cosine of pi over 6 is the square root of 3 over 2, and the cosine of pi over 4 is root 2 over 2. Then I'm subtracting the sine of pi over 6, which is 1 half, multiplied by the sine of pi over 4, which is root 2 over 2. Now let's do some algebra. Root 3 over 2 times root 2 over 2 is a root 6 over 4. Then 1 half times root 2 over 2 is a root 2 over 4, and these have a common denominator. So I can go ahead and combine these into 1 fraction, root 6 denominator. So I can go ahead and combine these into 1 fraction, root 6 minus root 2 over 4, and this is my final answer here. Thanks for watching and I'll see you in the next one.

6. Trigonometric Identities and More Equations

Sum and Difference Identities

Trigonometry6. Trigonometric Identities and More EquationsSum and Difference Identities

Multiple Choice

Find the exact value of the expression.
$\cos\frac{5\pi}{12}$

$\frac{\sqrt2-\sqrt6}{4}$

$\frac{\sqrt6-\sqrt2}{4}$

$\frac{\sqrt2-\sqrt6}{2}$

$\frac{\sqrt4}{4}$

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Find the exact value of the expression.﻿cos⁡5π12\cos\frac{5\pi}{12}cos125π​﻿

Find the exact value of the expression.
$\cos\frac{5\pi}{12}$