Problem 100 - Precalculus Essentials

Question

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\sin 2 x+\cos x=0$$

Step-by-Step Solution

Answer

The solution to the equation \(\sin 2x + \cos x = 0\) on the interval \([0,2\pi)\) are \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}\).

Step 1: Identify the transformation

Express the sine function in terms of cosine function. We recall the double angle identity for sine, \(\sin 2x = 2\sin x\cos x\). Thus, the given equation is now \(2\sin x\cos x + \cos x = 0\).

Step 2: Factor out common terms

The equation can be factored to reveal \(\cos x(2\sin x + 1) = 0\).

Step 3: Solve the factored equations

Setting each factor equal to zero gives \(\cos x = 0\) and \(2\sin x + 1 = 0\). The first equation yields \(x = \frac{\pi}{2}, \frac{3\pi}{2}\) within the interval \([0, 2\pi)\). The second equation gives \(\sin x = -\frac{1}{2}\), which yields \(x = \frac{7\pi}{6}, \frac{11\pi}{6}\) in \([0, 2\pi)\).

Step 4: Consolidate all solutions

Taking all solutions under consideration, we have four solutions for \(x\) within the given interval: \(\frac{\pi}{2}, \frac{3\pi}{2}, \frac{7\pi}{6}\) and \(\frac{11\pi}{6}\).

Key Concepts

Trigonometric Identities

When solving trigonometric equations, understanding trigonometric identities becomes crucial. These identities are like shortcuts that reveal relations between trigonometric functions. One well-known identity is the Pythagorean identity, which states:

• \( \sin^2 x + \cos^2 x = 1 \)

Another essential identity is the double angle identity. These transformations allow us to express functions like sine or cosine in terms of other angles, simplifying equations. The ability to spot and use these identities helps in manipulating equations to a simpler form. Whether it’s an angle sum or a double angle, knowing when and which identity to use is a key skill in trigonometry.

Double Angle Formula

The double angle formula is a valuable identity in trigonometry that helps relate angles that are twice that of another. For sine functions, the double angle formula is:

• \( \sin 2x = 2\sin x\cos x \)

In this particular exercise, we employed this identity to transform \( \sin 2x \) into terms containing \( \sin x \) and \( \cos x \). This transformation is essential as it allows us to convert the original equation into a more manageable format. Understanding and applying these formulas provides a powerful tool for tackling complex trigonometric problems, enabling quicker and more efficient solutions.

Factoring Equations

Factoring is a technique used in algebra and trigonometry to simplify expressions or equations. In this specific problem, once the double angle formula transformed the equation to \( 2\sin x\cos x + \cos x = 0 \), we factor out the common term, \( \cos x \). This changed our equation to:

• \( \cos x(2\sin x + 1) = 0 \)

Factoring allows us to split the equation into two simpler equations: \( \cos x = 0 \) and \( 2\sin x + 1 = 0 \). By breaking it down, we focus on solving each smaller equation separately, resulting in a set of solutions for \( x \). Factoring makes solving complex equations more straightforward by handling one piece at a time.

Sine and Cosine Functions

Sine and cosine functions are fundamental in trigonometry, representing periodic wave-like patterns. In solving trigonometric equations, these functions frequently appear because of their periodic properties. Cosine and sine functions have specific values at certain angles, which makes them predictable. For example, the cosine function has a zero value at angles \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) within the interval \([0, 2\pi)\). Similarly, sine has values such as \( \sin x = -\frac{1}{2} \) at angles \( \frac{7\pi}{6} \) and \( \frac{11\pi}{6} \). Understanding these values and their corresponding angles is crucial in solving trigonometric equations efficiently. These periodic properties allow us to predict and calculate solutions quickly.