Questions: Solve for the exact solutions in the interval [0,2 π). Separate solutions with a comma. If the equation has no solutions, respond with DNE. sin (5 x)=-sqrt(3)/2

Solution Steps

We start with the equation

\[ \sin(5x) = -\frac{\sqrt{3}}{2} \]

The sine function equals \(-\frac{\sqrt{3}}{2}\) at specific reference angles. The angles where this occurs in the unit circle are

\[ \frac{4\pi}{3} \quad \text{and} \quad \frac{5\pi}{3} \]

The general solutions for \(5x\) based on the reference angles are given by:

\[ 5x = \frac{4\pi}{3} + 2k\pi \quad \text{and} \quad 5x = \frac{5\pi}{3} + 2k\pi \]

where \(k\) is any integer.

Dividing each part of the equations by 5, we find:

\[ x = \frac{4\pi}{15} + \frac{2k\pi}{5} \quad \text{and} \quad x = \frac{5\pi}{15} + \frac{2k\pi}{5} \]

We need to find the specific solutions for \(k = 0\) and \(k = 1\) that fall within the interval \([0, 2\pi)\):

1. For \(k = 0\):

   • From \(x = \frac{4\pi}{15}\)
   • From \(x = \frac{5\pi}{15} = \frac{\pi}{3}\)

2. For \(k = 1\):

   • From \(x = \frac{4\pi}{15} + \frac{2\pi}{5} = \frac{4\pi}{15} + \frac{6\pi}{15} = \frac{10\pi}{15} = \frac{2\pi}{3}\)
   • From \(x = \frac{5\pi}{15} + \frac{2\pi}{5} = \frac{5\pi}{15} + \frac{6\pi}{15} = \frac{11\pi}{15}\)

Continuing this process for \(k = 2\):

• From \(x = \frac{4\pi}{15} + \frac{4\pi}{5} = \frac{4\pi}{15} + \frac{12\pi}{15} = \frac{16\pi}{15}\)
• From \(x = \frac{5\pi}{15} + \frac{4\pi}{5} = \frac{5\pi}{15} + \frac{12\pi}{15} = \frac{17\pi}{15}\)

Continuing for \(k = 3\):

• From \(x = \frac{4\pi}{15} + \frac{6\pi}{5} = \frac{4\pi}{15} + \frac{18\pi}{15} = \frac{22\pi}{15}\)
• From \(x = \frac{5\pi}{15} + \frac{6\pi}{5} = \frac{5\pi}{15} + \frac{18\pi}{15} = \frac{23\pi}{15}\)

Continuing for \(k = 4\):

• From \(x = \frac{4\pi}{15} + \frac{8\pi}{5} = \frac{4\pi}{15} + \frac{24\pi}{15} = \frac{28\pi}{15}\)
• From \(x = \frac{5\pi}{15} + \frac{8\pi}{5} = \frac{5\pi}{15} + \frac{24\pi}{15} = \frac{29\pi}{15}\)

The exact solutions in the interval \([0, 2\pi)\) are:

\[ \frac{4\pi}{15}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{11\pi}{15}, \frac{16\pi}{15}, \frac{17\pi}{15}, \frac{22\pi}{15}, \frac{23\pi}{15}, \frac{28\pi}{15}, \frac{29\pi}{15} \]

Final Answer