Find the exact value of the expression. cos(160[])cos(40[])+sin(160[])sin(40)?

Question

1. -sqrt[]3
2.-1/2
3. -sqrt[]3/2
4. -2

Answer 1

It looks like you made a typo. I'm guessing that the real question is an expansion of cos (160 degrees - 40 degrees). I.e., it's cos (120 degrees).

That said, one of the basic things you learn in trig is how to quickly calculate any of the six trig functions of any angle that's a multiple of 30 degrees (also known as pi/6 radians) or of 45 degrees (also known as pi/4 radians). For example, cos(t) = -cos (180 degrees - t), for any t. So cos (120 degrees) = -cos (60 degrees).

And you'd better know what cos(60 degrees) is. :)

Answer 2

It is impossible to write down an exact answer for these angles. It has to do with the nonconstructibility of 20 and 40 degree angles. There are exact formulas available for trigonometric functions of any multiple of 3 degrees, and there are formulas for a few others (see Gauss's formula), but for most angles the trig functions can only be expressed approximately, in terms of infinite series or complex integrals.