SOLUTION: Need help - Solve cos(x/4) = 1/2 in interval (-6pi,4pi).

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Question 1048008: Need help - Solve cos(x/4) = 1/2 in interval (-6pi,4pi).
Answer by Edwin McCravy(20063) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28x%2F4%29%22%22=%22%221%2F2

x%2F4%22%22=%22%22matrix%281%2C3%2Cpi%2F3%2B2pi%2Ak%2C%22%3B%22%2C5pi%2F3%2B2pi%2Ak%29

We'll take each one individually.

x%2F4%22%22=%22%22pi%2F3%2B2pi%2Ak

Multiply through by 4

x%22%22=%22%224pi%2F3%2B8pi%2Ak

in interval (-6pi,4pi)}}}

-6pi%22%22%3C%22%224pi%2F3%2B8pi%2Ak%22%22%3C%22%224pi

Multiply all three sides by 3

-18pi%22%22%3C%22%224pi%2B24pi%2Ak%22%22%3C%22%2212pi

Subtract 4pi from all three sides:

-14pi%22%22%3C%22%2224pi%2Ak%22%22%3C%22%228pi

Divide all three sides by 24pi

%28-14pi%29%2F%2824pi%29%22%22%3C%22%22%2824pi%2Ak%29%2F%2824pi%29%22%22%3C%22%228pi%2F%2824pi%29

%28-7%29%2F%2812%29%22%22%3C%22%22k%22%22%3C%22%221%2F3

The only integer between those values is k = 0.

So one solution is 

x%22%22=%22%224pi%2F3%2B8pi%2A0

x%22%22=%22%224pi%2F3

------------

For the other one,

x%2F4%22%22=%22%225pi%2F3%2B2pi%2Ak

Multiply through by 4

x%22%22=%22%2220pi%2F3%2B8pi%2Ak

in interval (-6pi,4pi)}}}

-6pi%22%22%3C%22%2220pi%2F3%2B8pi%2Ak%22%22%3C%22%224pi

Multiply all three sides by 3

-18pi%22%22%3C%22%2220pi%2B24pi%2Ak%22%22%3C%22%2212pi

Subtract 20pi from all three sides:

-38pi%22%22%3C%22%2224pi%2Ak%22%22%3C%22%22-8pi

Divide all three sides by 24pi

%28-38pi%29%2F%2824pi%29%22%22%3C%22%22%2824pi%2Ak%29%2F%2824pi%29%22%22%3C%22%22-8pi%2F%2824pi%29

%28-19%29%2F%2812%29%22%22%3C%22%22k%22%22%3C%22%22-1%2F3

%22-1.583...%22%22%22%3C%22%22k%22%22%3C%22%22%22-0.333...%22

The only integer between those values is k = -1.

So the other solution is 

x%22%22=%22%2220pi%2F3%2B8pi%2Ak

x%22%22=%22%2220pi%2F3%2B8pi%28-1%29

x%22%22=%22%2220pi%2F3-8pi

x%22%22=%22%2220pi%2F3-24pi%2F3

x%22%22=%22%22-4pi%2F3

x%22%22=%22%22-pi%2F3

Answer:  4pi%2F3 and -4pi%2F3

Edwin