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Understanding double angle formulas is a critical component of trigonometry and precalculus. These formulas are used to simplify the expressions involving trigonometric functions of double angles, such as \(2x\), \(2\theta\), or in this case, \(6x\), where the original angle is \(3x\).

There are three primary double angle formulas that you should be familiar with:

• \( \sin(2A) = 2\sin(A)\cos(A) \)
• \( \cos(2A) = \cos^2(A) - \sin^2(A) \) or alternate forms using Pythagorean identities
• \( \tan(2A) = \frac{2\tan(A)}{1 - \tan^2(A)} \)

In the exercise, we are verifying the identity involving the sine double angle formula. The key to using these formulas effectively lies in recognizing when to apply them and how they can simplify complex trigonometric expressions.

Trigonometric functions such as sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)) are fundamental in the study of angles and their relationships with triangle sides. In precalculus, these functions help in solving problems related to various phenomena including waves, oscillations, and circular motions.

Each trigonometric function has a specific definition and can be represented on the unit circle or via right-angled triangles. For example, the sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse. Similarly, cosine is the ratio of the adjacent side to the hypotenuse, while tangent is the ratio of the opposite side to the adjacent side.

Knowing how to manipulate these functions using algebraic identities and formulas is essential to verifying identities as seen in the exercise above, where we combine the sine and cosine of the same angle to verify a double angle identity.

Precalculus problem solving involves a combination of analytical thinking, algebraic manipulation, and grasp of trigonometric identities. It sets the stage for calculus by introducing concepts of limits, continuity, and complex functions. To solve problems effectively, one must first understand the question and identify which mathematical principles are relevant.

Following a step-by-step approach can greatly aid in solving problems:

1. Read the problem carefully and determine the objective.
2. Select the appropriate formulas or theorems.
3. Perform algebraic manipulations to simplify expressions.
4. Verify your result by substituting values or using alternative methods.

For instance, in this exercise, we apply these steps by identifying the double angle formula for sine, applying it to the given expression, and verifying that the simplified form matches the given identity. Meticulous work during these steps helps avoid common pitfalls and ensures a valid proof or solution.