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



Answer to a math 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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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 of 2\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}.

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