anonymous 5 years ago cos(3x) +0.5 = 0 I wan to understand how to solve it.

1. anonymous

anyone?

2. anonymous

x = .6981317008 this is the answer btw i know it, but i dont know how to get to it

3. anonymous

Okay, you can bring the 0.5 to the right hand side and you're left with cos(3x) = -0.5. You know, or should be familiar with the fact that the arc-cosine of 0.5 is 60 degrees, or pi/3 radians. So, because the value is negative, it's in the second quadrant and you can say that it's 120 degrees, or 2*pi/3 radians. Now, you're left with 3x = 2*pi/3. Divide by 3, and see the values that come out, and you should get about 0.69813... radians.

4. anonymous

I assume 0 < x < 2pi cos(3x)=-(1/2) since we know that cos(pi/3) = 1/2 and cosine is negative in the 2nd and 3rd quadrants, we have cos(3x) = cos(2pi/3) or cos (4pi/3) or cos(2pi + 2pi/3) or cos(2pi + 4pi/3) or cos(4pi + 2pi/3) or cos(4pi + 4pi/3) = cos(2pi/3) or cos(4pi/3) or cos(8pi/3) or cos(10pi/3) or cos(14pi/3) or cos(16pi/3) Thus 3x = 2pi/3 or 4pi/3 or 8pi/3 or 20pi/3 or 14pi/3 or 16pi/3. This gives your values x between 0 and 2pi

5. anonymous

$\cos(3x)=-0.5$ so 3x is either in the 2nd or 3rd quadrant, $3x=2\pi/3+2n \pi$ or $3x=4\pi/3+2n \pi, n=0,1,2,...$

6. anonymous

first of all let's write down the equation in terms of cos ^_^ and you'll get : $\cos(3x) = -0.5$ now you want to find x here, if you've noticed 3x acts as the angle in this question, so simply imagine it as theta and find the angle by doing the following: $3x = \cos^{-1} -0.5$ $x = [\cos^{-1} (-0.5)]/3 = 2\pi/9 = 40$ Correct me if I'm wrong :)

7. amistre64

cos(3x) = -1/2 ; the definition of cosine consists of parts of a triangle. It will be the "leg" that is right next to it; divided by the slanted (hypotenuse); in this case, its leg=1 and slant=2 leg^2 + (opposite leg)^2 = slanted^2 -1^2 + y^2 = 2^2 1 + y^2 = 4 y^2 = 3 y=sqrt(3) or y=-sqrt(3) since the leg is (-1) we take that to be the Q2. as it turns out; 120 degrees has a cosine of -1/2 in Q2.... gotta go....

8. anonymous

I think he wanted to solve for x, right? ._.

9. amistre64

3x = 120 x=120/3 x=40

10. amistre64

180+60 = 240 3x = 240 as well 3x=240 x=240/3 x=80...

11. amistre64

figure out which one = .69813.... lol.... neither one :)