91Ó°ÊÓ

The sine function, commonly noted as \( \sin \), is a fundamental aspect of trigonometry. It is used to determine the ratio of the opposite side to the hypotenuse in a right triangle. The unit circle provides an insightful way to understand sine, as it describes points along the circle's perimeter with a radius of 1.

In relation to angles, the sine function takes an angle as input and outputs a value between -1 and 1. This cyclic nature of the sine function is defined by its periodicity, reiterating every \(2\pi\) radians. Important angles that students should memorize are those whose sine values are well-known, such as \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\). For instance, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), which is the key value used in our original exercise.

• Memorizing unit circle values helps in solving equations quickly.
• Understanding periodicity aids in finding all possible solutions for sine equations.

When solving a trigonometric equation involving the sine function, like \( \sin(10\theta) = \frac{\sqrt{3}}{2} \), determining the angle solutions is crucial. We must recognize the angles that produce a specific sine value, here both \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \) satisfy this condition. These are known as angle solutions because they direct the principal values where the sine is equal to \( \frac{\sqrt{3}}{2} \).

Since the sine function repeats every \(2\pi\), additional solutions exist by adding multiples of \(2\pi\) - this concept is called periodicity.

• Principal values give initial angles satisfying the sine equation.
• Periodicity extends these initial solutions to include all possible angles.

The general solution in the context of trigonometric equations like \( \sin(10\theta) = \frac{\sqrt{3}}{2} \) is how we express all possible solutions. We start by identifying angle solutions, such as \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \), and extend these using the formula for general solutions: \[10\theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad 10\theta = \frac{2\pi}{3} + 2k\pi\]where \( k \) is any integer. This accounts for the infinite nature of the sine wave's periodicity.

To isolate \( \theta \), divide through by 10:\[\theta = \frac{\pi}{30} + \frac{k\pi}{5} \quad \text{and} \quad \theta = \frac{\pi}{15} + \frac{k\pi}{5}\]This compact form gives all possible angles \( \theta \) that satisfy the original equation.

• Always express general solutions with variable \( k \) to denote all possible angles.
• Note how dividing by constants adapts periodic solutions for different coefficients.