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°

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12. Trigonometric Identities

Sum and Difference Identities

Precalculus

12. Trigonometric Identities

Sum and Difference Identities

Struggling with Precalculus?

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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°)cos(20°) + sin(80°)sin(20°) resembles the cosine addition formula: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).

Identify the angles in the expression: A = 80° and B = 20°.

Apply the cosine addition formula: cos(80° + 20°) = cos(100°).

Understand that cos(100°) can be simplified using the identity cos(180° - θ) = -cos(θ), which gives cos(100°) = -cos(80°).

Conclude that the expression simplifies to -cos(80°), and further simplification or evaluation can be done based on the context or additional information provided.