# Solve cos 6x=(1/2) for principal values in degree

• MHB
• blake1
In summary, the equation cos 6x = (1/2) has solutions for x at intervals of pi/18 radians or 10 degrees, within the range of 0 to 2pi.
blake1
cos 6x=(1/2)

blake said:
cos 6x=(1/2)

note $\cos{\theta} = \dfrac{1}{2}$ at $\theta = \dfrac{\pi}{3} \text{ and } \dfrac{5\pi}{3}$

$0 \le x < 2\pi \implies 0 \le 6x < 12\pi$

$\cos(6x) = \dfrac{1}{2} \implies 6x = \dfrac{\pi}{3} \, , \, \dfrac{5\pi}{3} \, , \, \dfrac{7\pi}{3} \, , \, \dfrac{11\pi}{3} \, , \, \dfrac{13\pi}{3} \, , \, \dfrac{17\pi}{3}\, , \, \dfrac{19\pi}{3} \, , \, \dfrac{23\pi}{3} \, , \, \dfrac{25\pi}{3} \, , \, \dfrac{29\pi}{3} \, , \, \dfrac{31\pi}{3} \, , \, \dfrac{35\pi}{3}$

$x = \dfrac{\pi}{18} \, , \, \dfrac{5\pi}{18} \, , \, \dfrac{7\pi}{18} \, , \, \dfrac{11\pi}{18} \, , \, \dfrac{13\pi}{18} \, , \, \dfrac{17\pi}{18}\, , \, \dfrac{19\pi}{18} \, , \, \dfrac{23\pi}{18} \, , \, \dfrac{25\pi}{18} \, , \, \dfrac{29\pi}{18} \, , \, \dfrac{31\pi}{18} \, , \, \dfrac{35\pi}{18}$

Skeeter's answer is, of course, in radians. To get the answer in degrees remember that $$\pi$$ radians is 180 degrees. That is, $$\frac{180}{\pi}= 1$$ so $$\frac{\pi}{18}$$ radians is the same as $$\frac{\pi}{18}\frac{180}{\pi}= 10$$ degrees.

## 1. What is the principal value of cos 6x when it is equal to 1/2?

The principal value of cos 6x when it is equal to 1/2 is approximately 60 degrees or π/3 radians.

## 2. How do you solve cos 6x=(1/2) for principal values in degrees?

To solve cos 6x=(1/2) for principal values in degrees, you can use the inverse cosine function or the unit circle. Simply take the inverse cosine of 1/2, which is 60 degrees, and then divide by 6 to get the principal value of x as 10 degrees.

## 3. Is there more than one principal value for cos 6x=(1/2) in degrees?

Yes, there are multiple principal values for cos 6x=(1/2) in degrees. This is because the cosine function is periodic and has a repeating pattern. The other possible principal values are 180 degrees or π radians plus the initial principal value of 10 degrees, resulting in 190 degrees or 7π/6 radians, and 360 degrees or 2π radians plus the initial principal value of 10 degrees, resulting in 370 degrees or 13π/6 radians.

## 4. Can you solve cos 6x=(1/2) for principal values in radians?

Yes, you can solve cos 6x=(1/2) for principal values in radians using the same method as solving for degrees. The only difference is, you will use radians instead of degrees in your calculations.

## 5. What is the general solution for cos 6x=(1/2) in degrees?

The general solution for cos 6x=(1/2) in degrees is x = 10 + 360n, where n is any integer. This is because the cosine function repeats every 360 degrees, so adding or subtracting a multiple of 360 will result in another solution.

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