91Ó°ÊÓ

First, let's understand trigonometric functions. The basic trigonometric functions are sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}). These functions relate the angles of a triangle to the lengths of its sides in a right-angled triangle. They are fundamental in studying periodic phenomena like waves. In our equation, we are dealing with \(\text{sin}(\theta)\). To solve \(\text{sin}^2(\theta)-1=0\), knowing about these functions is crucial.

The unit circle is a powerful tool to understand trigonometric functions. It is a circle with a radius of 1 centered at the origin (0,0) on a coordinate plane. Any point on the unit circle has coordinates \((\text{cos}(\theta), \text{sin}(\theta))\) where \(\theta\) is the angle made with the positive x-axis. When you need to find where \(\text{sin}(\theta)=1\) or \(\text{sin}(\theta)=-1\), you look at the y-coordinates on the unit circle. For example, \(\text{sin}(\theta)=1\) at \(\theta = \frac{\fpi}{2}\) and \(\text{sin}(\theta)=-1\) at \(\theta = \frac{3\fpi}{2}\).

Interval notation is used to describe sets of numbers between two endpoints. In this exercise, we're focusing on the interval \(0 \theta<2 \text{\fpi}\).
This means that \(θ\) starts at 0 and goes up to, but does not include, \(2 \text{\fpi}\). We use interval notation to precisely specify the range of our solutions. In this case, you're looking for angles \(θ\) within \(0 \theta<2 \text{\fpi}\) where \(\text{sin}(\theta) = \text{-1}\) \text{or} \(\text{sin}(\theta) = \text{1}\).

Solving trigonometric equations involves isolating the trigonometric function and then finding the angle that satisfies the equation within the specified interval. Here’s our step-by-step method used:
Start by rewriting the equation: \(\text{sin}^2(\theta) = 1\).
Next, take the square root of both sides gives \(\text{sin}(\theta)= \text{1 \text{or}} \text{-1}\).
Then, determine in which angles in the interval \(0 \theta<2 \text{\fpi}\) these solutions occur.
Lastly, interpret these solutions with the unit circle to find the exact angles: \(\text{sin}(\theta) = 1\) at \(\theta = \frac{\fect{\text{\fpi}}{2}}\) and \(\text{sin}(\theta) = -1\) \frac{3\fect{\text{\fpi}{2}}.$$.
By combining these steps, you find all solutions in the given interval.