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The quadratic formula is a powerful tool for solving equations of the form \( ax^2 + bx + c = 0 \). This kind of equation is known as a quadratic equation. By using the quadratic formula, you can find the values of \( x \) that satisfy the equation by following a consistent process. Here's a rundown of what it involves and how it's used in trigonometric equations.

• Identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation.
• Substitute these values into the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Carefully solve for \( x \) by calculating the square root and division.

This formula is critical for finding solutions such as \( \sin x \) when faced with quadratic trigonometric problems. It's also important to simplify throughout the process to make calculations easier. Remember, any number that can result from this process is potentially a solution, but it should be evaluated in the context of a trigonometric problem, as seen with \( \sin x = 1 \) and \( \sin x = \frac{3}{4} \) in the original problem.

The sine function is one of the fundamental trigonometric functions often used to describe the relationships in triangles, especially right-angled ones. For angles, it is defined as the ratio of the opposite side to the hypotenuse in a right triangle. However, in the context of equations, it helps in understanding periodic behaviors and oscillations.

• The function is an input-output mechanism where an angle (in radians or degrees) is given as input and the ratio is the output.
• Sine values range between -1 and 1, which is why solutions to trigonometric equations are expected within this range.
• Sine has its maximum value at \( x = \frac{\pi}{2} \) where \( \sin x = 1 \).
• It is symmetric and can provide positive values in both the first and second quadrants.

In the exercise, the sine function is central to moving from \( \sin x = \{1, \frac{3}{4}\} \) to finding the angles that produce these values. Especially note \( \sin x = \frac{3}{4}\), where specific quadrant knowledge helps determine other angles in the given interval.

Inverse trigonometric functions are essential for finding angles when given a sine value, as seen with \( \sin x = \frac{3}{4} \). When you have the result of a sine function and need to determine the angle, you use the inverse sine function, denoted as \( \text{arcsin} \) or \( \sin^{-1} \). This concept allows us to reverse-engineer from the trigonometric values back to the angle.

• An inverse trigonometric function receives a value between -1 and 1 and produces an angle whose sine, cosine, or tangent is that value.
• \( \text{arcsin} \left(x\right) \) produces an angle in radians.
• The inverse function is useful for finding primary solutions; however, it can also be used to find secondary angles by considering the symmetry of the sine function.

In our example, calculating \( \text{arcsin} \left( \frac{3}{4} \right) \) gives an initial angle of approximately 0.848. Recognizing that sine is positive in the second quadrant as well, we add \( \pi - \text{arcsin} \left( \frac{3}{4} \right) \) to find another valid solution within our specified range. This demonstrates the usefulness of inverse trigonometric functions in solving complex equations efficiently.