91Ó°ÊÓ

To begin solving trigonometric equations, understanding trigonometric identities is essential. Trigonometric identities are equations involving trigonometric functions that are true for any value of the included variables. They help simplify and solve trigonometric equations. Let's see how we used one in this exercise.

We started with the trigonometric identity \(\text{cos}(A - B) = \text{cos}A \text{cos}B + \text{sin}A \text{sin}B\). This identity lets us rewrite \(\text{cos}(x - \frac{\text{pi}}{2})\) in a more straightforward form.

When we plug in \(\text{A} = x\) and \(\text{B} = \frac{\text{pi}}{2}\), we find that \(\text{cos}(x - \frac{\text{pi}}{2})\) simplifies to \(\text{sin}(x)\). Knowing which identity to use and how to apply it efficiently is key in trigonometry.

For ease, always remember:

• Basic angle sum and difference identities.
• Reciprocal identities, like \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\).
• Pythagorean identities, such as \(\text{sin}^2(x) + \text{cos}^2(x) = 1\).

These identities are the foundation for solving more complex trigonometric equations.

The sine function, \(\text{sin}(x)\), is a periodic function that oscillates between -1 and 1. Understanding its properties helps us solve trigonometric equations like \(\text{sin}(x) = \frac{\text{sqrt}{3}}{2}\) efficiently.

The sine function repeats every \(2\text{pi}\) and intersects the x-axis at multiples of \(\text{pi}\). Here, we found the angles where \(\text{sin}(x) = \frac{\text{sqrt}{3}}{2}\). These angles are:

• \(\frac{\text{pi}}{3}\)
• \(\frac{2\text{pi}}{3}\)

Because the sine function is periodic, these solutions can be written in the general forms: \(\frac{\text{pi}}{3} + 2k\text{pi}\) and \(\frac{2\text{pi}}{3} + 2k\text{pi}\), where \(k\) is any integer. This generalization helps us find all possible solutions to the sine equation. Analyze these intervals through graphs or unit circles for a better visual understanding.

Interval analysis helps narrow down the solutions of trigonometric equations within a specific range. In this exercise, we were given the interval \(-\text{pi} < x < \text{pi}\).

We must check our general solutions to determine which fall within this interval.

• For \(\frac{\text{pi}}{3}\), check if it lies between \(-\text{pi} < x < \text{pi}\). It does, as \(\frac{\text{pi}}{3}\) is approximately 1.047, which fits within -3.14 to 3.14.
• For \(\frac{2\text{pi}}{3}\), do the same. This value is about 2.094, also within the interval.

By systematically analyzing these intervals, you ensure that the solutions you find are indeed valid. Always:

• Convert the general solution to the specific interval.
• Use a unit circle or diagrams for more complex intervals.
• Double-check boundary values to include or exclude them if they are strict inequalities.

The final valid solutions within the given interval are \(\frac{\text{pi}}{3}\) and \(\frac{2\text{pi}}{3}\).