91Ó°ÊÓ

In trigonometry, converting between degrees and radians is a crucial skill. Degrees and radians are both units used to measure angles. Understanding the relationship between these units allows us to work seamlessly with angles in various applications. Here's how it works:

• There are 360 degrees in a circle and \( 2\pi \) radians in a circle.
• This establishes the conversion factors: \( 180^{\circ} = \pi \) radians.

To convert degrees to radians, multiply the degree measurement by \( \frac{\pi}{180} \). For example, to convert \( 55.1^{\circ} \) to radians, we compute \( 55.1 \times \frac{\pi}{180} \approx 0.96 \) radians.
Conversely, to convert from radians to degrees, multiply the radian measurement by \( \frac{180}{\pi} \). For example, converting \( 2.18 \) radians to degrees, we get \( 2.18 \times \frac{180}{\pi} \approx 124.9^{\circ} \). Understanding these conversions is key when finding solutions in trigonometric equations over specific intervals.

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). In our exercise, the trigonometric equation was transformed into a quadratic form using \sin substitution. Here, the standard quadratic formula is applied:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Where \( a, b, \) and \( c \) are coefficients from the quadratic equation.

For the equation \( 9y^2 - 6y - 1 = 0 \), the coefficients are \( a = 9 \), \( b = -6 \), and \( c = -1 \). Following the formula, the solution gives values of \( y \) that relate back to \sin x.
The discriminant, \( b^2 - 4ac \), indicates the nature of the solutions. A positive discriminant, as in this case \( 72 \), results in two distinct real solutions which can be used to find the angles in our problem.

The sine function is fundamental in trigonometry and relates an angle to the ratio of the opposite side to the hypotenuse in a right-angled triangle. This function is defined for all real numbers and varies between -1 and 1. In solving trigonometric equations, understanding the sine function is necessary when interpreting solutions:

• Given \sin x = \( y \), find \( x \) using inverse sine functions.
• The basic form of the sine function repeats every \( 2\pi \) radians, indicating a cycle.

In the exercise, after solving the quadratic equation in terms of \sin x, we are left with valid solutions \( \sin x = 0.81 \) and \( \sin x = -0.47 \). To find the corresponding angles \( x \), use \( x = \sin^{-1}(0.81) \) and for \( \sin x = -0.47 \), find the negative angle first and adjust within the given interval to get valid solutions.

When finding solutions to trigonometric equations, they often need to be restricted to a specific interval. In this case, the problem specifies intervals

• from \([0, 2\pi)\) in radians
• and \([0^{\circ}, 360^{\circ})\) in degrees.

This means finding all possible angles within these boundaries where the equation holds true.
For \sin x = 0.81, the primary solution angle in degrees is around \( 55.1^{\circ} \). To find other solutions within \([0, 360)\), consider the supplementary angle \( 180^{\circ} - 55.1^{\circ} = 124.9^{\circ} \).
For \sin x = -0.47, we initially find the angle \(-28.1^{\circ}\). Since this is outside \([0, 360)\), we adjust by adding \(360^{\circ}\) to obtain \( 331.9^{\circ} \), and the other valid solution at \( 180^{\circ} + 28.1^{\circ} = 208.1^{\circ} \).
Finally, we translate these degree measures into radians, ensuring all solutions neatly fit within \([0, 2\pi)\).