91Ó°ÊÓ

Trigonometric identities are equations involving trigonometric functions that hold true for all values of angles. One key identity used in this problem is \[ \cos^2 x = 1 - \sin^2 x. \]This identity helps us transform the original equation involving cosine into an equation involving only sine. The reason for this transformation is to simplify the equation into a single trigonometric function, making it easier to solve.Trigonometric identities are essential tools in trigonometry, providing a way to relate different trigonometric functions. You might often use identities to simplify complex equations, solve trigonometric expressions, or prove other mathematical statements. Understanding these identities helps in simplifying equations and in finding solutions that might not be immediately obvious.

A quadratic equation is an equation of the form:\[ ax^2 + bx + c = 0, \]where \(a\), \(b\), and \(c\) are constants. In this exercise, after substituting \(y = \sin x\), our trigonometric equation becomes a quadratic equation: \[ 2y^2 - 5y + 2 = 0. \]Solving quadratic equations can be done using different methods:

• Factoring
• Using the quadratic formula
• Completing the square

For this specific problem, the equation is solved using factoring to find the roots \(y = 1\) and \(y = \frac{1}{2}\). Each root corresponds to a potential solution for \(\sin x\). Quadratic equations often arise in trigonometry when transforming and solving equations, making them an important concept to master.

Finding trigonometric solutions involves determining the angle values that satisfy given trigonometric equations. After re-substituting the values from the solved quadratic equation, we have:

• \(\sin x = 1\)
• \(\sin x = \frac{1}{2}\)

To find \(x\) for \(\sin x = 1\), the solution is:
\[ x = \frac{\pi}{2} + 2k\pi, \]where \(k\) is any integer, representing the angle \(90^\circ\) plus any full rotations.For \(\sin x = \frac{1}{2}\), the solutions are:

• \(x = \frac{\pi}{6} + 2k\pi\)
• \(x = \frac{5\pi}{6} + 2k\pi\)

These represent angles \(30^\circ\) and \(150^\circ\) plus any full rotations. Trigonometric solutions require understanding the periodic nature of sine and cosine functions and their specific values at notable angles.