Question

Determine a general formula (or formulas) for the solution to the following equation. Then, determine the specific solutions (if any) on the interval [0,2π). cos30=0

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Maude

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To determine the general formula for the solution to the equation \cos(30^\circ) = 0 , we can start by converting the given angle from degrees to radians since trigonometric functions typically work with radians.

We know that\pi radians is equal to 180^\circ , therefore, 30^\circ is equal to \frac{\pi}{6} radians.

Now, we can use the inverse cosine function\arccos to find the solution:

\arccos(0) = \frac{\pi}{6} + n\pi , where n is an integer.

This is because the cosine function has a period of2\pi , and \cos(\theta) = 0 occurs at \theta = \frac{\pi}{2} + n\pi , where n is an integer.

Now, let's determine the specific solutions on the interval[0,2\pi) :

For the given equation, we have:

\frac{\pi}{6} + n\pi for n = 0 ,

\frac{\pi}{6} + n\pi for n = 1 .

Therefore, the specific solutions on the interval[0,2\pi) are:

\frac{\pi}{6} and \frac{7\pi}{6} .

Answer: The general formula for the solution to the equation\cos(30) = 0 is \frac{\pi}{6} + n\pi , where n is an integer. The specific solutions on the interval [0,2\pi) are \frac{\pi}{6} and \frac{7\pi}{6} .

We know that

Now, we can use the inverse cosine function

This is because the cosine function has a period of

Now, let's determine the specific solutions on the interval

For the given equation, we have:

Therefore, the specific solutions on the interval

Answer: The general formula for the solution to the equation

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