Questions: Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." 5√3sin x + 5 = 0 x =

Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." 5√3sin x + 5 = 0 x =
Transcript text: Question 2 of 15, Step 1 of 1 0/15 Correct Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." $5\sqrt{3}\sin x + 5 = 0$ Answer: How to enter your answer (opens in new window): Keyboard shortcuts Enter your answer in radians, as an exact answer when possible. Multiple solutions should be separated by commas. Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used. x = ☐ No Solution Submit Answer © 2025 Hawkes Learning
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Solution

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Solution Steps

Step 1: Isolate the Trigonometric Function

The given equation is:

\[ 5\sqrt{3}\sin x + 5 = 0 \]

First, isolate the sine function by subtracting 5 from both sides:

\[ 5\sqrt{3}\sin x = -5 \]

Step 2: Solve for \(\sin x\)

Divide both sides by \(5\sqrt{3}\) to solve for \(\sin x\):

\[ \sin x = \frac{-5}{5\sqrt{3}} = -\frac{1}{\sqrt{3}} \]

Rationalize the denominator:

\[ \sin x = -\frac{\sqrt{3}}{3} \]

Step 3: Find the General Solution

The sine function is negative in the third and fourth quadrants. The reference angle \(\theta\) for \(\sin \theta = \frac{\sqrt{3}}{3}\) is \(\frac{\pi}{6}\).

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

  • Third quadrant: \(x = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\)
  • Fourth quadrant: \(x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\)

Final Answer

The solutions to the equation \(5\sqrt{3}\sin x + 5 = 0\) in the interval \([0, 2\pi)\) are:

\[ \boxed{x = \frac{7\pi}{6}, \frac{11\pi}{6}} \]

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