Find the exact value of the expression.cos5π12\cos\frac{5\pi}{12}cos125π

6−24\frac{\sqrt6-\sqrt2}{4}46−2

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.

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

6−24\frac{\sqrt6-\sqrt2}{4}46−2

2−64\frac{\sqrt2-\sqrt6}{4}42−6

2−62\frac{\sqrt2-\sqrt6}{2}22−6

6. Trigonometric Identities and More Equations

Sum and Difference Identities

Trigonometry

6. Trigonometric Identities and More Equations

Sum and Difference Identities

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

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

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

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

$\frac{\sqrt4}{4}$

Guida verificata passo dopo passo

First, recognize that $ \cos\frac{5\pi}{12} $ is an angle that can be expressed in terms of known angles. We can use the angle sum or difference identities to find its exact value.

Express $ \frac{5\pi}{12} $ in terms of angles whose cosine values are known. Notice that $ \frac{5\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} $. This allows us to use the cosine difference identity: $ \cos(a - b) = \cos a \cos b + \sin a \sin b $.

Substitute $ a = \frac{\pi}{3} $ and $ b = \frac{\pi}{4} $ into the identity: $ \cos\left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \cos\frac{\pi}{3} \cos\frac{\pi}{4} + \sin\frac{\pi}{3} \sin\frac{\pi}{4} $.

Calculate the known values: $ \cos\frac{\pi}{3} = \frac{1}{2} $, $ \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2} $, $ \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} $, and $ \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} $.

Substitute these values into the expression: $ \cos\frac{5\pi}{12} = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $. Simplify the expression to find the exact value.