Question

How do I find the exact value of cos 4pi/3?

Answer 1

Wolfram alpha :P

Answer 2

\[\color{blue}{ Cos \frac{4π}{3} }\]
\[\color{red}{ we~~~know:~~~~~~~~~π=180° }\]
\[\color{blue}{ Cos\frac{4 \times 180}{3} } \]
\[\color{blue}{ Cos(240) } \]
\[\color{red} { identity:~~~~ \cos(A-B)=\cos (A) \cos (B) - \sin (A) \sin(B) } \]\[\color{blue}{ Cos(240) } =\color{blue}{ Cos(180+60) =Cos(180)~Cos(60)-Sin(180)~Sin(60)}\]
\[\color{red} { exact~~~values:~~~~~Cos(180)=0,~~~~Sin(180)=1,~~~~ Cos(60)=0.5~~~~ Sin(60)= \frac{\sqrt{3} }{2} } \]

Answer 3

\[\color{blue}{ Cos(180+60)=(0 \times 0.5)~~-~~(1 \times \frac{\sqrt{3} }{2} ) = -\frac{\sqrt{3} }{2} }\]

Answer 4

There is the entire process.

Answer 5

(4 pi)/3 is 240 degrees. Use a calculator. Or you could look at the unit circle. There are some good videos on that.

Answer 6

No, no calculator, there is an exact value, just like I showed above.

Answer 7

I'm not sure if it's because I'm using a MAC, but here is what I see above:

[\color{blue}{ Cos \frac{4π}{3} }\] \[\color{red}{ we~~~know:~~~~~~~~~π=180° }\] \[\color{blue}{ Cos\frac{4 \times 180}{3} } \] \[\color{blue}{ Cos(240) } \] \[\color{red} { identity:~~~~ \cos(A-B)=\cos (A) \cos (B) - \sin (A) \sin(B) } \]\[\color{blue}{ Cos(240) } =\color{blue}{ Cos(180+60) =Cos(180)~Cos(60)-Sin(180)~Sin(60)}\] \[\color{red} { exact~~~values:~~~~~Cos(180)=0,~~~~Sin(180)=1,~~~~ Cos(60)=0.5~~~~ Sin(60)= \fra

Answer 8

That is what I see above as well, its a little hard to read through. But I am NOT supposed to use a calculator.

Answer 9

There are several ways you can do this. By far the easiest method of solving it would be to write cos(4pi/3) as a sum of two "known" values.

In this case we can write cos(4pi/3) as cos (pi + pi/3)=cos(pi)*cos(pi/3)-sin(pi)*sin(pi/3)= (-1)*(1/2) - 0 (because sin(pi)=0). So the answer is -1/2.

Answer 10

yeah, where I said " exact values: " in red, Sin(60)= sqrt{3} / 2

Answer 11

Try to learn about the unit circle and the two special triangles (30-60-90) and (45-45-90). They can be very helpful for questions like this.

Answer 12

If you look at the unit circle and where the radius is for an angle of 240 degrees, do you see why the x value and y value would both be negative? It's because of which quadrant you're in.

Answer 13

jam333 THANK YOU SOOO MUCH for drawing it out for me, I've been having trouble doing that...just cann't get the hang of it. I'm also still trying to figure the unit circle out. Hopefully I will soon.

Answer 14

PatrickJMT has a good unit circle demo on youtube.