Question

Exact value of cos(3pi+pi/3)

Answer 1

-1/2

Answer 2

we know that \(\cos(x)=\cos(x+2\pi)\) hence we can add/subtract integer multiples of \(2\pi\) without affecting the value... if we subtract \(2\pi\) we get \(3\pi+\pi/3-2\pi=\pi+\pi/3\):

$$\cos(3\pi+\pi/3)=\cos(\pi+\pi/3)$$

Answer 3

now, in case you're not familiar with what \(\cos(4\pi/3)\) is, we can use identities to tackle the problem:$$\cos(\pi+\pi/3)=\cos(\pi)\cos(\pi/3)-\sin(\pi)\sin(\pi/3)$$since \(\cos(a+b)=\cos a\cos b-\sin a\sin b\)

Answer 4

we know \(\cos\pi=-1\) while \(\sin \pi=0\) so the second term disappears and we have:$$\cos(\pi)\cos(\pi/3)=-\cos(\pi/3)$$

Answer 5

now we just recall from the unit circle \(\cos(\pi/3)=1/2\) thus our answer is \(-\cos(\pi/3)=-1/2\)