Question

Solve \(\sqrt{3} \cos x \csc x=-2 \cos x\) for \(x\) over the domain \(\mathbf{0} \leq x<2 \pi\).

Short answer

\( x = \frac{4 \pi}{3} \) and \( x = \frac{5 \pi}{3} \)

Step-by-step solution

- Simplify the given equation

Start by simplifying the given equation: \ \ \( \sqrt{3} \cos x \csc x = -2 \cos x \) \ Consider that \(\csc x = \frac{1}{\sin x}\) and substitute it: \ \ \( \sqrt{3} \cos x \frac{1}{\sin x} = -2 \cos x \) \ Simplify further: \ \ \( \frac{\backslashsqrt{3} \cos x}{\sin x} = -2 \cos x \)

- Divide both sides by \( \cos x \)

Divide both sides of the equation by \( \cos x \) (assuming \( \cos x \eq 0 \)): \ \ \( \frac{\backslashsqrt{3}}{\sin x} = -2 \) \ This simplifies to: \ \ \( \frac{\backslashsqrt{3}}{\sin x} = -2 \)

- Solve for \( \sin x \)

Rearrange the equation to solve for \( \sin x \): \ \ \( \sin x = \frac{\backslash-sqrt{3}}{2} \)

- Identify possible values for \( x \) within the domain

Now identify the values of \( x \) within the domain \( \{0 \leq x < 2 \pi \} \) where \( \sin x = \frac{-\sqrt{3}}{2} \). This occurs at: \ \ \( x = \frac{4 \pi}{3} \) and \( x = \frac{5 \pi}{3} \)

Key concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that make the equation true. These values are generally the angles at which the trigonometric functions reach specific results. This particular problem required solving the equation: \[ \frac{\backslashsqrt{3}}{\backslashsin x} = -2 \]. Here's a general approach for solving trigonometric equations:

• Simplify the equation using algebraic manipulations.
• Leverage trigonometric identities to express the equation in a simpler form. (For example, using \( \backslashcsc x = \frac{1}{\backslashsin x} \))
• Isolate the trigonometric function.
• Find the general solutions for this function.
• Identify specific solutions within the given domain.

Breaking the process into these steps can help you tackle even the most complex trigonometric equations. Starting with simplification and substitution as in the given equation not only makes it easier to solve but also reduces errors.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable for which both sides are defined. In this exercise, recognizing and using the identity \( \backslashcsc x = \frac{1}{\backslashsin x} \) was crucial. Here are some key identities to keep in mind:

• \( \backslashsin ^2 x + \backslashcos ^2 x = 1 \)
• \( \backslashtan x = \frac{\backslashsin x}{\backslashcos x} \)
• \( \backslashcsc x = \frac{1}{\backslashsin x} \)
• \( \backslashsec x = \frac{1}{\backslashcos x} \)
• \( \backslashcot x = \frac{\backslashcos x}{\backslashsin x} \)

By mastering these identities, you'll be able to simplify and manipulate trigonometric equations more effectively. The given problem demonstrated the use of \( \backslashcsc x \) to transform the original equation to a more workable form.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It's a circle with radius 1 centered at the origin of a coordinate plane. Understanding the unit circle is essential for solving equations involving trigonometric functions, as it provides the values of functions like \( \backslashsin \) and \( \backslashcos \) at different angles. Here's why it's important in this problem:

• For \( \backslashsin x = \frac{-\backslashsqrt{3}}{2} \), you need to find where this value occurs on the unit circle.
• On the unit circle, \( \backslashsin x \) corresponds to the y-coordinate of a point.
• \( \backslashsin x = \frac{-\backslashsqrt{3}}{2} \) occurs at specific standard angles.

From the unit circle, one can see that \( \backslashsin x \) equals \( \frac{-\backslashsqrt{3}}{2} \) at angles \( \frac{4 \backslashpi}{3} \) and \( \frac{5 \backslashpi}{3} \). Knowing the unit circle structure allows for quick identification of these angles, which complete the solution. This is why familiarity with the unit circle is so valuable.