91Ó°ÊÓ

The sine function, often written as \(\text{sin} \alpha\), is a fundamental concept in trigonometry. It connects an angle \(\text{α}\) with the ratio of sides in a right triangle. Specifically, the sine of an angle is the length of the side opposite the angle divided by the length of the hypotenuse (the longest side of the triangle).
The formula looks like this: \(\text{sin} \alpha = \frac{\text{opposite}}{\text{hypotenuse}}\).
The values of the sine function range between -1 and 1 for all angles. This means that for any angle \(\text{α}\), \(\text{sin} \alpha\) will always be somewhere between \(-1\textrm{ and }1\).
This property is crucial in solving trigonometric equations. Understanding the behavior and range of the sine function is foundational for finding angle solutions.

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. This is a powerful tool in trigonometry because it helps to visualize and understand the values of sine, cosine, and other trigonometric functions for different angles.
Every point on the unit circle has coordinates \((\text{x}, \text{y})\), where \(\text{x}\) represents \(\text{cos} \alpha\) and \(\text{y}\) represents \(\text{sin} \alpha\).
Whenever you have an angle, you can place it on the unit circle. For instance, at \({270^\text{°}} \text{ or } \frac{3\text{\textpi}}{2} \text{ radians}\), the coordinates are (0, -1). This means that \({\text{sin } 270^\text{°} = -1}\).
This circle helps to understand how the sine function behaves and repeats every 360 degrees (or 2pi radians). Hence, it is easier to find all possible angle solutions.

Solving trigonometric equations involves finding all the angles that satisfy the given equation. For the equation \(\text{sin} \alpha = -1\), knowing the behavior and key properties of the sine function helps. On the unit circle, \(\text{sin} \alpha\) equals \(-1\) exactly at \({270^\text{°}} \text{ or } \frac{3\text{\textpi}}{2} \text{ radians}\).
However, since the sine function is periodic with a period of 360 degrees, every multiple of 360 degrees added to 270 will also be a solution.
Thus, the general formula for all solutions to the equation \(\text{sin} \alpha = -1\) is: \({270^\text{°} + 360^\text{°}k}\), where \(\text{k}\) is any integer. This means:

• \(\text{k}=0\) \rightarrow \alpha = 270^\text{°}\
• \(\text{k}=1\) \rightarrow \alpha = 630^\text{°} = 270^\text{°} + 360^\text{°}\
• \(\text{k}=-1\) \rightarrow \alpha = -90^\text{°} = 270^\text{°} - 360^\text{°}\

By adjusting the value of \(\text{k}\), you can find all angles that satisfy the initial equation.