Problem 161 - Precalculus Essentials

Question

Solve each equation on the interval \([0,2 \pi)\) Do not use a calculator. $$\sin 3 x+\sin x+\cos x=0$$

Step-by-Step Solution

Answer

As the final equation contains cubic term, solving this equation without a calculator becomes quite difficult and time-consuming. Numerical methods are typically used to solve it.

Step 1: Express sin(3x) using trigonometric identities

The formula for sin(3x) can be expressed using the sum of angles formula for sine as \(\sin 3x=3\sin x - 4\sin^{3}x\). So, the equation becomes \(3\sin x - 4\sin^{3}x + \sin x + \cos x = 0\).

Step 2: Combine like terms

Combine like terms to simplify the equation. Rewrite the equation as \(4\sin x + \cos x - 4\sin^{3}x = 0\).

Step 3: Converting cos(x) to sin(x)

The cosine expression can be written in terms of sine using the Pythagorean identity \(\cos x = \sqrt{1 - \sin^{2}x}\). After this step, the equation will contain only sine, which will simplify the resolution. So, the new equation becomes \(4\sin x + \sqrt{1 - \sin^{2}x} - 4\sin^{3}x = 0\). Note that since x is in the range [0, 2π], cos(x) is positive, which justifies the positive root for the square root.

Step 4: Solve for sin(x)

In this step, rearrange terms and solve for \(\sin x\). This yields a cubic equation which may require further techniques to solve, including the factor theorem, synthetic division, or the rational root theorem. Remember to check your solutions against the given interval [0, 2π].

Key Concepts

Sine and Cosine Identities

Sine and cosine identities are fundamental in solving trigonometric equations. They relate the sine and cosine functions at different angles, helping to simplify complex expressions. One useful identity is the triple-angle formula for sine:

• \( \sin 3x = 3\sin x - 4\sin^3 x \)

This identity expresses \( \sin 3x \) in terms of \( \sin x \), making problems like "\( \sin 3 x+\sin x+\cos x=0 \)" more approachable. Knowing how to manipulate and apply these identities can significantly reduce problem complexity, leading to easier solutions without a calculator.
By expressing the sine and cosine terms in uniform trigonometric terms, you can often reduce mixed trigonometric equations into simpler forms, easing the path to finding solutions within specified intervals.

Pythagorean Identities

Pythagorean identities are key tools in converting between sine and cosine in an equation. The most commonly used identity is:

• \( \sin^2 x + \cos^2 x = 1 \)

In the given problem, the cosine term was converted into a function of sine using the identity derived from rearranging the above:

• \( \cos x = \sqrt{1 - \sin^2 x} \)

This conversion is extremely useful when you're trying to solve equations involving mixed trigonometric terms.
Keep in mind the range or interval of the angle, as this affects which root (positive or negative) of the square root is selected.
In the interval \([0, 2\pi)\), ensuring \( \cos x \) remains positive simplifies the solving process by maintaining consistency in the solutions.

Cubic Equations

Cubic equations, such as those derived from trigonometric equations, involve terms up to the third degree. Solving them requires understanding specific algebraic techniques. In this trigonometric context, when rearranging terms to form a cubic equation, the equation often looks like:

• \( 4\sin x + \sqrt{1 - \sin^2 x} - 4\sin^3 x = 0 \)

The solutions are the values of \( \sin x \) that satisfy this expression. Techniques like factor theorem, synthetic division, or the rational root theorem are used to break down these equations into solvable parts.
Once you have possible solutions, check each one against the interval provided, in this case, \([0, 2\pi)\), to ensure they are valid within the given boundary.

Interval Solutions

Interval solutions are about finding the correct values of \( x \) within a specified range. For trigonometric equations, the solution is often limited to an interval, such as \([0, 2\pi)\). This ensures you find all correct angles for which the equation holds true. When solving, you need to:

• Consider the periodic nature of trigonometric functions.
• Ensure the solutions are within the given interval.
• Verify each potential solution as some may not satisfy initial constraints.

Trigonometric solutions often repeat outside of given intervals; thus, adhering to the interval ensures a complete but non-redundant set of solutions.
After finding potential values, double-check against the conditions set by the problem to ensure all found solutions are valid.