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The sine function is a fundamental concept in trigonometry, often represented as \( \sin x \). It is key in understanding various geometric shapes and waves in mathematics and physics. - **Relationship with Right Triangles:** In a right triangle, the sine of an angle \( x \) is the ratio of the length of the side opposite to the angle \( x \) and the hypotenuse.- **Sine Wave:** If you've seen graphs of the sine function, you'll notice it creates a smooth, wave-like curve, known as a sine wave.In terms of unit circle representation:- **Unit Circle:** The value of the sine function at any angle \( x \) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle.- **Range:** The sine values are always within the interval [\(-1, 1\)].When working with specific intervals, like \( \left(\frac{\pi}{2}, \pi\right) \), it is important to determine the sign of sin. In this particular interval, the sine function is positive, helping us to solve expressions such as \( \sin u \). By finding \( \sin u = \cfrac{2}{\sqrt{5}} \) from our original formula, we affirm that within our interval, the sine value will always remain positive.

The tangent function, denoted as \( \tan x \), is another fundamental trigonometric function. It often expresses the ratio of the sine and cosine of an angle. Here are some important aspects:- **Definition:** \( \tan x = \cfrac{\sin x}{\cos x} \). This definition relates directly to the values used in the problem.- **Behavior in Intervals:** In our exercise, \( \tan u = -2 \) and \( \tan v = -3 \). This highlights that the tangent function can be negative depending on the angle's position on the unit circle.For right-angle triangles:- **Relation to Angles:** The tangent of an angle is the ratio of the opposite side to the adjacent side.Moreover, within the interval \( \left(\frac{\pi}{2}, \pi\right) \), tangent values are negative. This property is crucial in identifying and confirming the behavior of \( \sin u \), using identities such as \(1 + \tan^2 u = \cfrac{1}{\cos^2 u} \) to solve for \( \cos^2 u \) and eventually \( \sin u \).

Interval notation is a concise way to describe a range of values, often used in solutions to define precise ranges for angles and values in trigonometry. In our problem, we used interval notation to define the location of \( u \) and \( v \):- **Format:** For any interval, such as \( (a, b) \), the values are those for which the variable is greater than \( a \) and less than \( b \).- **Applying to Trigonometric Functions:** The interval \( \left(\frac{\pi}{2}, \pi\right) \) tells us that the angles lie in the second quadrant, where certain trigonometric functions take specific values.Why this is important:- **Quadrants and Trigonometry:** Knowledge about which trigonometric functions are positive or negative in particular quadrants empowers us to evaluate expressions correctly. For example, in this interval, we know that \( \sin u \) is positive.When solving trigonometric problems, using interval notation helps to narrow down the possibilities and ensure the solutions are valid within given bounds, as we derived from \( \sin u = \cfrac{2}{\sqrt{5}} \). It helps you handle these functions in a structured, recognizable format.