Find the exact value of the expression. cos 80 ° cos 20 ° + sin 80 ° sin 20 ° \cos80\degree\cos20\degree+\sin80\degree\sin20\degree cos 80° cos 20° + sin 80° sin 20°

Here we're asked to find the exact value of the expression the cosine of 80 degrees times the cosine of 20 degrees, plus the sine of 80 degrees, times the sine of 20 degrees. Now I don't know these trig values off the top of my head, but I do know that this looks really familiar. Because looking at my cosine formula here, I know that I have a cosine times a cosine plus a sine times a sine. Now because I have that, this tells me that this is really just equal to the cosine of 80 degrees minus 20 degrees. Now, if you're ever unsure and you can't get to that right away remember that you can always expand this out using your formula to make sure that you get back to that same expression that you started with. Now from here, if I take 80 degrees minus 20 degrees, I end up with the cosine of 60 degrees, which is an angle that I already know. The cosine of 60 degrees is equal to 1 half, and that's my final answer here. The cosine of 80 degrees times the cosine of 20 degrees plus the sine of 80 degrees times the sine of 20 degrees is equal to 1 half. Thanks for watching and I'll see you in the next one.

Find the exact value of the expression. cos ⁡ 80 ° cos ⁡ 20 ° + sin ⁡ 80 ° sin ⁡ 20 ° \cos80\degree\cos20\degree+\sin80\degree\sin20\degree cos 80° cos 20° + sin 80° sin 20°

3 2 \frac{\sqrt3}{2} 2 3 <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">

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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Multiple Choice

Find the exact value of the expression.
$\cos80\degree\cos20\degree+\sin80\degree\sin20\degree$

$-\frac12$

$\frac12$

$\frac{\sqrt3}{2}$

Verified step by step guidance

Recognize that the expression $ \cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ $ resembles the cosine addition formula: $ \cos(a - b) = \cos a \cos b + \sin a \sin b $.

Identify the angles in the expression: $ a = 80^\circ $ and $ b = 20^\circ $.

Apply the cosine addition formula: $ \cos(80^\circ - 20^\circ) = \cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ $.

Calculate the angle difference: $ 80^\circ - 20^\circ = 60^\circ $.

Conclude that the expression simplifies to $ \cos 60^\circ $, which is a known trigonometric value.

6:14m

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Struggling with Trigonometry?

Find the exact value of the expression.cos80°cos20°+sin80°sin20°\cos80\degree\cos20\degree+\sin80\degree\sin20\degreecos80°cos20°+sin80°sin20°

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Sum and Difference Identities practice set

Find the exact value of the expression.
$\cos80\degree\cos20\degree+\sin80\degree\sin20\degree$