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 one 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 one half. Thanks 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">

12. Trigonometric Functions

Trigonometric Identities

Business Calculus

12. Trigonometric Functions

Trigonometric Identities

Struggling with Business Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

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 given expression cos(80°)cos(20°) + sin(80°)sin(20°) resembles the trigonometric identity for cos(a - b), which is cos(a - b) = cos(a)cos(b) + sin(a)sin(b).

Identify the values of a and b in the expression. Here, a = 80° and b = 20°.

Substitute these values into the identity: cos(80° - 20°) = cos(80°)cos(20°) + sin(80°)sin(20°).

Simplify the expression inside the cosine function: 80° - 20° = 60°.

Conclude that the expression simplifies to cos(60°). Use the known value of cos(60°) = 1/2 to verify the result if needed.

6:19m

Watch next

Master Even and Odd Identities with a bite sized video explanation from Patrick

Start learning

Trigonometric Identities practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice

Struggling with Business Calculus?

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

Watch next

Trigonometric Identities practice set

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