91Ó°ÊÓ

The angle sum identity for the sine function is a fundamental trigonometric formula. It allows us to express the sine of a sum of two angles in terms of the sine and cosine of the individual angles. Specifically, this identity states:
\[ \text{sin}(a + b) = \text{sin}(a)\text{cos}(b) + \text{cos}(a)\text{sin}(b) \]
This identity is crucial because it lets us transform more complex trigonometric equations into simpler forms that are easier to solve.
In our exercise, by recognizing the left-hand side of the given equation as the sine of a difference, we apply a variant of the angle sum identity:
\[ \text{sin}(a - b) = \text{sin}(a)\text{cos}(b) - \text{cos}(a)\text{sin}(b) \]
Here, letting \( a = \frac{\text{Ï€}}{6} \) and \( b = x \), we utilize the identity to rewrite the given equation.

The sine function, denoted as \( \text{sin}(x) \), is one of the primary functions in trigonometry. It is defined for all real numbers, but it repeats its values in a periodic manner with a period of \( 2\text{Ï€} \). This property means \( \text{sin}(x + 2\text{Ï€}) = \text{sin}(x) \) for any angle \( x \).
Starting from the unit circle perspective, \( \text{sin}(x) \) represents the y-coordinate of a point on the arc corresponding to the angle \( x \).
The sine function ranges from -1 to 1 and has some key points such as:

• \( \text{sin}(0) = 0 \)
• \( \text{sin}(\frac{\text{Ï€}}{2}) = 1 \)
• \( \text{sin}(\text{Ï€}) = 0 \)
• \( \text{sin}(\frac{3\text{Ï€}}{2}) = -1 \)

In our exercise, we use the sine function’s properties to find solutions to equations involving sine values set to specific numbers.

Trigonometric intervals specify the range or domain in which we look for angle solutions. In our given problem, the intervals are \([0, 2Ï€)\), which means we need to find all solutions for the variable within one complete revolution of the unit circle.

• \(0 \leq x < 2Ï€\)

This interval restricts solutions to one period of the sine function. When calculating solutions, we should ensure they lie within this range by adding or subtracting multiples of \(2Ï€\) if necessary.
The interval \([0, 2Ï€)\) is chosen often because it represents all possible angles without repeating values, making it easier to interpret solutions in a single cycle of trigonometric functions.

Solving trigonometric equations involves finding all angles that satisfy a given equation. Here are the steps generally followed:

• **Rewriting the equation** using trigonometric identities to make it simpler.
• **Isolating the trigonometric function**, for example, getting \( \text{sin}(x) = c \).
• **Finding the general solutions** to the equation. For sine, this would involve solving \( x = \text{arcsin}(c) + 2k\text{Ï€} \) or \( x = \text{Ï€} - \text{arcsin}(c) + 2k\text{Ï€} \).
• **Considering the interval** restrictions and ensuring that all solutions lie within the specified range.

In our exercise, once we have simplified the equation:
\[ \text{sin}(\frac{Ï€}{6} - x) = -\frac{1}{2} \]
We find the angles where sine equals \(-\frac{1}{2}\). Solving \( \frac{Ï€}{6} - x = -\frac{Ï€}{6} + 2k\text{Ï€} \) and \( \frac{Ï€}{6} - x = \frac{7Ï€}{6} + 2k\text{Ï€} \). Adjusting for the interval \([0, 2\text{Ï€})\), we find the final solutions \( x = \frac{Ï€}{3} \) and \( x = \text{Ï€} \).