91Ó°ÊÓ

The arcsine function, denoted as \( \text{sin}^{-1}x \) or \( \text{arcsin} \, x \), is the inverse of the sine function. When you use the arcsine function on a value, you are looking for an angle whose sine is that value. For example, if \( \text{sin} \theta = x \), then \( \text{arcsin} \, x = \theta \). It’s important to note that the arcsine function has a restricted range of \( -\frac{\text{pi}}{2} \) to \( \frac{\text{pi}}{2} \). This means the result of an arcsine operation will always be an angle between these two values.

In our exercise, we started with the equation \( 4 \, \text{sin}^{-1} x = \text{pi} \). Understanding the properties of the arcsine function helps in solving this. First, we need to isolate the \( \text{sin}^{-1} x \) term by dividing both sides by 4. Then, we solve for \( x \) by finding the sine of both sides, converting the equation to \( x = \text{sin} \left( \frac{\text{pi}}{4} \right) \).

Solving trigonometric equations involves isolating the trigonometric function and then solving for the variable. Here are the steps generally followed:

• Isolate the trigonometric function: Rearrange the equation so that the trigonometric function is alone on one side. In our example, we turned \( 4 \, \text{sin}^{-1} x = \text{pi} \) into \( \text{sin}^{-1} x = \frac{\text{pi}}{4} \).
• Apply the inverse trigonometric function: Use the corresponding inverse trigonometric function to find the variable. Applying \( \text{sin} \) to both sides of \( \text{sin}^{-1} x = \frac{\text{pi}}{4} \) gave us \( x = \text{sin} \left( \frac{\text{pi}}{4} \right) \).

These steps are essential in solving any trigonometric equation and can be applied to different functions. Just remember to follow them carefully, and you'll reach the solution step by step.

Standard angles in trigonometry, such as \( 0, \frac{\text{pi}}{6}, \frac{\text{pi}}{4}, \frac{\text{pi}}{3}, \text{and} \frac{\text{pi}}{2} \) are commonly used and their sine values are often memorized. Knowing these angles makes solving trigonometrical equations quicker. Here's a quick reference:

• \( \text{sin}(0) = 0 \)
• \( \text{sin} \left( \frac{\text{pi}}{6} \right) = \frac{1}{2} \)
• \( \text{sin} \left( \frac{\text{pi}}{4} \right) = \frac{\text{sqrt}(2)}{2} \)
• \( \text{sin} \left( \frac{\text{pi}}{3} \right) = \frac{\text{sqrt}(3)}{2} \)
• \( \text{sin} \left( \frac{\text{pi}}{2} \right) = 1 \)

In our example equation, we needed to find \( \text{sin} \left( \frac{\text{pi}}{4} \right) \), which we know is \( \frac{\text{sqrt}(2)}{2} \). This is why the solution leads to \( x = \frac{\text{sqrt}(2)}{2} \). Understanding these standard values helps solve problems faster and with greater accuracy.