91Ó°ÊÓ

To solve this exercise, we utilized the sine angle addition formula. This identity is essential in trigonometry because it allows us to simplify expressions involving the sine of the sum of two angles. The formula is written as: \ \[ \ \ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \] This means that if you have sine plus cosine terms multiplied together, you might be able to simplify them to a single sine term representing the sum of the two angles.

An exact trigonometric value refers to the specific number that a trigonometric function equals for certain angles. These values are derived from the unit circle and are well-known for standard angles like 0, 30, 45, 60, and 90 degrees (or 0, \[ \frac{\pi}{6} \], \[ \frac{\pi}{4} \], \[ \frac{\pi}{3} \], and \[ \frac{\pi}{2} \] radians, respectively). For instance: \ \begin{align*} \ \sin \frac{\pi}{2} &= 1 \ \cos \frac{\pi}{2} &= 0 \ \sin \frac{\pi}{4} &= \frac{\sqrt{2}}{2} \ \cos \frac{\pi}{4} &= \frac{\sqrt{2}}{2} \ \end{align*} In this problem, we used the fact that \[ \sin \frac{\pi}{2} \] equals 1. Knowing these exact values is crucial for quickly solving trigonometric problems.

Simplifying trigonometric expressions often involves using trigonometric identities. These include the angle addition and subtraction formulas, double-angle formulas, and more. To simplify the given expression: \ \[ \sin \frac{\pi}{5} \cos \frac{3 \pi}{10}+\cos \frac{\pi}{5} \sin \frac{3 \pi}{10} \] We used the sine angle addition identity to rewrite it as \[ \sin \left( \frac{\pi}{5} + \frac{3 \pi}{10} \right) \]. Once simplified further by adding the angles, the problem became \[ \sin \frac{\pi}{2} = 1 \]. Utilizing identities transforms complex expressions into something simpler and easier to evaluate. Breaking down the steps and substitutions ensures you don't miss any critical detail.