Problem 99 - Algebra and Trigonometry

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. $$ \sin 2 x+\sin x=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \( \sin 2 x+\sin x=0 \) on the interval \([0,2 \pi)\) are \( x = 0, \pi, 2/3 \pi, 4/3 \pi \)

Step 1: Apply the Identity

Here the double-angle identity \( \sin 2x = 2\sin x \cos x \) is applied to the equation. Substituting this gives: \( 2 \sin x \cos x + \sin x = 0 \)

Step 2: Factorise the Equation

The equation can be rewritten by factoring out common term \( \sin x \): \( \sin x (2 \cos x + 1) = 0 \)

Step 3: Solve for the Values of x

This gives two equations when equal to zero: \( \sin x = 0 \) and \( 2\cos x + 1 = 0\). Solve for \( x \) for each equation.

Step 4: Find the Solutions for \( \sin x = 0 \)

One solution arises from the equation \( \sin x = 0 \), which gives \( x = 0 \) and \( x = \pi \)

Step 5: Find the Solutions for \( 2\cos x + 1 = 0 \)

The other solutions come from \( 2\cos x + 1 = 0 \). Isolating \( \cos x \) we get \( \cos x = -1/2 \), which gives the solutions \( x = 2/3\pi \) and \( x = 4/3\pi \)

Key Concepts

Double-Angle Identities

When solving trigonometric equations, double-angle identities are powerful tools for simplifying expressions involving angles that are twice the size of an angle in the known functions. The identity for sine, specifically, relates the sine of a double angle to the product of sine and cosine of the original angle.

For example, the identity \[ \sin(2x) = 2\sin(x)\cos(x) \]is crucial in solving equations like \( \sin(2x) + \sin(x) = 0 \).By applying this identity, we can transform the double angle into forms involving single angles, which we can more easily solve by factoring or by applying standard trigonometric identities.

The keywords here are 'transforming' and 'simplifying'. Remember that using double-angle identities efficiently cuts complexity in half, setting you up for a more straightforward path to finding solutions.

Factoring Trigonometric Expressions

Factoring is a critical algebraic skill, especially useful in trigonometry for breaking down complex expressions into simpler, more manageable ones.

In the provided example, \( 2\sin(x)\cos(x) + \sin(x) = 0 \),we notice that \( \sin(x) \) is common to both terms.By factoring it out, we get \( \sin(x)(2\cos(x) + 1) = 0 \).This is a significant step towards solving trigonometric equations, as it leads us to a point where we can use the Zero-Product Property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero.

Once factored, individual terms can be examined to isolate solutions. So, when you're faced with a trigonometric equation that looks complicated, look for common factors to simplify the problem to something more familiar.

Finding Trigonometric Solutions

The goal when solving trigonometric equations is to find all the angle measurements that make the equation true within a given interval. In the case of the problem \( \sin(x)(2\cos(x) + 1) = 0 \), after factoring, we're left with two separate equations that need solving: \( \sin(x) = 0 \) and \( 2\cos(x) + 1 = 0 \).For \( \sin(x) = 0 \), we know that sine is zero at \( x = 0 \) and \( x = \pi \) within the interval \([0, 2\pi)\).

For \( 2\cos(x) + 1 = 0 \), we isolate \( \cos(x) \) to find that \( \cos(x) = -1/2 \), a value corresponding to specific angles on the unit circle.These must be calculated or looked up in trigonometric tables to find that the solutions are \( x = \frac{2}{3}\pi \) and \( x = \frac{4}{3}\pi \).

In summary, proper factoring and application of identities will bring you to a stage where you can easily find all the trigonometric solutions within the desired interval using known trigonometric values and properties.