91Ó°ÊÓ

In the exercise, we are working with a trigonometric equation that can be transformed into a quadratic form. The original equation is \( 2 \sin^2 \theta + (2-\sqrt{3}) \sin \theta - \sqrt{3} = 0 \). Recognizing that this is a quadratic equation in terms of \( \sin \theta \) is the first step.
A quadratic equation has the form \( ax^2 + bx + c = 0 \) and can be solved using the quadratic formula:

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.
• The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In our equation, we let \( \sin \theta = y \) to fit into the quadratic formula, resulting in \( a = 2 \), \( b = 2-\sqrt{3} \), and \( c = -\sqrt{3} \). This substitution simplifies solving trigonometric equations by treating them like traditional algebraic quadratic equations.

Once we find a solution for \( y = \sin \theta \), to determine \( \theta \), we must use inverse trigonometric functions. When \( y = \sin \theta \approx 0.366 \), to find \( \theta \), we use the inverse sine function:
\( \theta = \sin^{-1}(0.366) \).
Inverse trigonometric functions help us retrieve the angle given a specific sine, cosine, or tangent value.

• These functions include \( \sin^{-1} \), \( \cos^{-1} \), and \( \tan^{-1} \).
• The principal value of \( \sin^{-1} \) lies between \(-90^\circ\) and \(90^\circ\).

However, since the sine function is positive in both the first and second quadrants, we need to consider both corresponding angles. So, besides the principal angle \( \theta \approx 21.45^\circ \), we also find \( \theta \approx 180^\circ - 21.45^\circ = 158.55^\circ \). This comprehensive understanding allows for the complete set of solutions when using inverse trigonometric functions.

The general solution of a trigonometric equation accounts for all possible angles that satisfy the equation.
For the trigonometric equation \( 2 \sin^2 \theta + (2-\sqrt{3}) \sin \theta - \sqrt{3} = 0 \), after finding \( \theta \approx 21.45^\circ \) and \( 158.55^\circ \), we develop the general solution.
Trigonometric functions are periodic, meaning they repeat their values every specified interval (360 degrees for sine and cosine).

• The general solution in degrees is expressed as \( \theta = 21.45^\circ + 360^\circ n \) and \( \theta = 158.55^\circ + 360^\circ n \) where \( n \) is any integer.
• Each \( n \) increment represents one full rotation of the circle, bringing you back to an equivalent angle.

By including the variable \( n \), the general solution addresses the infinite nature of trigonometric solutions, conveying all possibilities within a periodic framework.