Problem 244 - Precalculus

Question

For the following exercises, find the exact value. $$ \sin ^{-1}\left(\frac{\sqrt{3}}{2}\right) $$

Step-by-Step Solution

Answer

\(\frac{\pi}{3}\)

Step 1: Understand the Problem

We need to find the angle whose sine value equals \( \frac{\sqrt{3}}{2} \). This is represented by \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \), which is the same as asking for the inverse sine or arcsine of \( \frac{\sqrt{3}}{2} \).

Step 2: Recall Sine Values of Common Angles

Recall that in a unit circle or common angle chart, \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \). Because of this, \( \frac{\pi}{3} \) is a candidate for solving \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \).

Step 3: Confirm the Quadrant

The value of \( \sin^{-1}(x) \) has to be in the range \([\frac{-\pi}{2}, \frac{\pi}{2}]\), the first and fourth quadrants of the unit circle. \( \frac{\pi}{3} \) falls within this range (first quadrant), confirming it as the solution.

Step 4: Solution Verification

Verify that \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) to ensure correctness. Since this relationship holds true, \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \) is the correct solution.

Key Concepts

Understanding Arcsine

The arcsine function, represented as \( \sin^{-1}(x) \), is an important inverse trigonometric function. It helps us find the angle whose sine value is \( x \). Unlike the sine function, which takes an angle and returns a value, the arcsine does the opposite.

When we explore the arcsine function, we're actually looking for an angle \( \theta \) such that \( \sin(\theta) = x \). The range of the arcsine function is restricted to \([\frac{-\pi}{2}, \frac{\pi}{2}]\). This restriction ensures that we get a unique angle for each valid input.

It's important to note that arcsine can only take values in the range from \(-1\) to \(1\) because these are the possible sine values of angles. Understanding this function is key to solving problems related to inverse trigonometric functions.

The Role of the Unit Circle

The unit circle is a fundamental tool in understanding trigonometric functions, including the arcsine. A unit circle is a circle with radius \(1\) centered at the origin of a coordinate plane. In the context of trigonometry, it allows us to easily visualize angles and their corresponding sine, cosine, and tangent values.

In a unit circle, when we refer to sine, we're looking at the vertical coordinate (y-value) of a point on the circle. For an angle \( \theta \) from the positive x-axis, the y-coordinate represents \( \sin(\theta) \).

Understanding the unit circle helps us easily identify sine values for different angles. For instance, on the unit circle, the angle \( \frac{\pi}{3} \) has a sine value of \( \frac{\sqrt{3}}{2} \). This makes interpreting the arcsine much more straightforward, as we see which angle matches our sine value in the correct quadrant.

Common Sine Values and Their Angles

Memorizing the sine values of common angles can drastically simplify trigonometry problems. Especially when dealing with inverse functions like arcsine.

There are some key angles whose sine values frequently appear in exercises:

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

By knowing these, you can quickly compute the arcsine of any common sine value, as you understand which angle corresponds to that sine.

Remembering these specific values not only aids in speed but ensures accuracy in identifying the correct angles within their appropriate ranges in problems involving inverse trigonometric functions.