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The double angle formulas are a cornerstone in solving trigonometric equations, especially those involving periodic functions.
These formulas express trigonometric functions of double angles, such as \( 2x \), in terms of single-angle quantities like \( x \). For sine, the important one is \( \sin(2x) = 2\sin(x)\cos(x) \).
Understanding this relationship is crucial because it allows us to break down more complex expressions into simpler terms that are easier to solve. When faced with an equation like \( \sin 2x + \cos x = 0 \), applying the double angle formula helps to reveal underlying factors that can then be isolated to find specific angle solutions within a given interval.

Factoring is a powerful algebraic tool used to simplify trigonometric equations and make them more solvable. By grouping common factors, we can reduce complex trigonometric expressions into a product of simpler expressions.
In our example, we took the trigonometric equation \( 2\sin(x)\cos(x) + \cos(x) = 0 \) and factored out \( \cos(x) \), yielding \( \cos(x)(2\sin(x) + 1) = 0 \). Factoring lays the groundwork for the zero-product property that allows us to solve for the variable by setting each factor to zero separately.

The goal when solving trigonometric equations is to isolate the variable, in this case, \( x \), and solve for its values within the specified interval \( [0,2 \pi) \).
Once the equation is factored, we apply the zero-product property: if a product equals zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve each resulting equation separately.
This gives us two sets of equations: \( \cos(x) = 0 \) and \( 2\sin(x) + 1 = 0 \), which we can solve to find exact or approximate solutions for \( x \) within the given interval.

In trigonometry, it's often possible to find exact values for certain angles. These are based on the unit circle and well-known right triangles, such as the 45-45-90 or 30-60-90 triangles.
For the equation \( \cos(x) = 0 \), the solutions within the interval \( [0,2 \pi) \) are \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) because these are the angles where the cosine function equals zero on the unit circle. Similarly, the solutions to \( \sin(x) = -1/2 \) are \( x = \frac{7\pi}{6} \) and \( x = \frac{11\pi}{6} \), which correspond to the exact values of sine for those angles.
Recognizing these angles and their corresponding trigonometric values is crucial for solving equations and helps students find precise results rather than relying on decimal approximations.