If \(A=(\sin 2)(\sin 3)(\sin 5)\) then \(\begin{array}{llll}\text { (a) } a>0 & \text { (b) } A=0 & \text { (c) } A<0 & \text { (d) } A \geq 0\end{array}\)

The sine function is one of the fundamental aspects of trigonometry, a branch of mathematics that studies the relationships between the sides and angles of triangles. Specifically, the sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse (the longest side of the triangle).

When we are looking at the unit circle – a circle with a radius of one -- the sine function can also be understood as the y-coordinate of a point on the circle's circumference. This is particularly useful when dealing with angles that cannot be easily constructed in a right-angled triangle.

Most calculators or mathematical software can provide the sine values for different angles. However, it is also important to consider the context of the angle, such as which quadrant it lies in, to determine the sign (positive or negative) of the sine function. As in the exercise, if the angles are in the first quadrant, one can immediately deduce that their sine values will be positive without needing to compute the actual values.

First quadrant angles refer to angles that are greater than 0° but less than 90°. In the context of trigonometry and the unit circle, any angle in the first quadrant will have both a positive cosine and a positive sine value. This is due to their geometric representations on the coordinate plane, where both x and y coordinates are positive in the first quadrant.

In problems like the one presented, recognizing that all given angles are within the first quadrant simplifies calculations. You don't need to compute exact values; knowing they are positive suffices for solving many types of problems. As such, one of the improvements suggested for students would be to strengthen their understanding of the coordinate plane and how it intersects with trigonometric functions. This conceptual knowledge greatly simplifies problems that would otherwise require complex calculations.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable(s) involved. These identities are very useful in simplifying trigonometric expressions and solving trigonometric equations. Some of the most fundamental identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities.

Understanding these identities allows students to transform complex trigonometric expressions into simpler ones, often converting products into sums or differential into division. In the context of the given exercise, knowing the positivity of the sine function in the first quadrant and using the appropriate identities can help in finding the solution more efficiently.

A crucial piece of advice is to familiarize yourself with these identities early on. While the solved problem does not directly utilize complex identities, having a command of them can simplify more convoluted problems that students may encounter in their studies.