91Ó°ÊÓ

The sine function, represented as \(\text{sin}(x)\), is a fundamental topic in trigonometry. It's often used to describe the relationship between the angles and sides of a right triangle. Specifically, for any angle \(\theta\), the sine value is the ratio of the length of the opposite side to the hypotenuse in a right triangle. This relationship is written as: \(\text{sin}(θ) = \frac{\text{opposite}}{\text{hypotenuse}}\).

Additionally, the sine function is defined for all real numbers and produces values between \(-1\) and \(1\). This makes it a periodic function with a wave-like pattern that repeats every \(\text{2Ï€}\) radians (or 360 degrees).

In the context of solving the given exercise \(2 \text{sin}(x) + \text{√2} = 0\), we first need to isolate \(\text{sin}(x)\). Understanding the sine function and its behavior helps us manipulate and solve the equation correctly.

A periodic function is a function that repeats its values in regular intervals or periods. For the sine function, the period is \(2Ï€\). This means that the sine curve will repeat its pattern every \(2Ï€\) units.

To grasp this better, visualizing the sine function can be helpful. The primary cycle starts at zero, rises to \(1\) at \(\frac{Ï€}{2}\), descends to zero at \(\text{Ï€}\), falls to \(-1\) at \(\frac{3Ï€}{2}\), and returns to zero at \(2Ï€\). Then it repeats this cycle.

Understanding periodicity is crucial in solving trigonometric equations. For instance, when we found \(\text{sin}(x) = \frac{-\text{√2}}{2}\) in our exercise, we used the fact that sine is periodic to write the general solutions as: \(\text{x} = -\frac{\text{π}}{4} + 2kπ \text{ or } x = \frac{5π}{4} + 2kπ\), where \(k\) is any integer. This periodic pattern allows us to succinctly express all possible solutions to the sine equation.

Solving trigonometric equations involves finding all angles \(x\) that satisfy the given equation. Let's break down the key steps using our example \(2 \text{sin}(x) + \text{√2} = 0\).

1. **Isolate the Trigonometric Function:** Our first goal is to isolate \(\text{sin}(x)\). Subtract \(√2\) from both sides of the equation: \(2 \text{sin}(x) = -√2\).

2. **Solve for the Function:** To get \(\text{sin}(x)\) by itself, divide both sides by 2: \(\text{sin}(x) = \frac{-\text{√2}}{2}\).

3. **Identify Possible Angles:** Determine which angles have the sine value \(\frac{-\text{√2}}{2}\). From the unit circle and knowledge of sine values, we recognize that \(\text{sin}(x) = \frac{-\text{√2}}{2}\) at \(\text{x} = -\frac{\text{π}}{4}\) and \(\text{x} = \frac{5π}{4}\).

4. **Consider Periodicity:** Since sine is periodic with a period of \(2π\), the complete solutions must include all such values. Thus, the general solutions are: \(\text{x} = -\frac{π}{4} + 2kπ \text{ or } x = \frac{5π}{4} + 2kπ\), where \(k\) symbolizes any integer.

By methodically isolating, solving, and applying the periodic properties of the sine function, we can tackle a wide range of trigonometric equations successfully.