Problem 12 - Precalculus: A Concise Course

Question

Evaluate the expression without using a calculator. $$ \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

-π/4

Step 1: Apply the definition of arcsine

The arcsine function, represented as \( \sin^{-1}(x) \), is the inverse of the sine function. To evaluate \( \sin^{-1}(-\sqrt{2}/2) \), we want to find the angle \( \theta \) in the interval from \( -\pi/2 \) to \( \pi/2 \) such that \( \sin(\theta) = -\sqrt{2}/2 \).

Step 2: Connect with the unit circle

Considering the concept of the unit circle, we recall that the coordinates at any point on the unit circle are given by \( (\cos\theta, \sin\theta) \). As we are given \( \sin(\theta) = -\sqrt{2}/2 \), this corresponds to a point on the circle that is on negative y-axis. Given the restriction of the arcsine function, this corresponds to a point in the 4th quadrant which has the coordinates \( (\sqrt{2}/2, -\sqrt{2}/2) \). This is associated with an angle -π/4 or -45 degrees.

Step 3: Answer

Therefore, the value of \( \sin^{-1}(-\sqrt{2}/2) \) is -π/4.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It provides a geometric way to visualize angles and their corresponding trigonometric values. The circle has a radius of one and is centered at the origin of the coordinate plane.

• The x-coordinate of a point on the unit circle represents the cosine of the angle associated with that point.
• The y-coordinate represents the sine of the angle.
• Every point on the unit circle corresponds to an angle from 0 to 2π radians, or 0 to 360 degrees.

Angles on the unit circle can be either positive or negative, depending on their direction. Counterclockwise angles are positive, while clockwise angles are negative. Understanding these coordinates and their angle associations is key to solving inverse trigonometric problems. In the provided exercise, we find a familiar point with coordinates close to \((\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\) that helps determine the arc sine value.

Arcsine Function

The arcsine function, denoted as \( ext{sin}^{-1}(x)\), is the inverse operation of the sine function. It is designed to find the angle whose sine is \(x\).

• It effectively 'undoes' the sine function.
• The defined range for the arcsine function is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) to ensure that it is one-to-one.
• This restriction is crucial because it tells us where to look for the angle that matches a given sine value.

In the given problem, \( ext{sin}^{-1}(-\frac{\sqrt{2}}{2})\) needs the angle \(\theta\) such that \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) and \( ext{sin}(\theta) = -\frac{\sqrt{2}}{2}\). Knowing this, we identify that the angle fulfilling all conditions is \(-\frac{\pi}{4}\) or \(-45\) degrees.

Angle Evaluation

Angle evaluation involves finding a specific angle based on its trigonometric properties or evaluations. In this problem, we seek the angle whose sine value matches a specific negative value of \(-\frac{\sqrt{2}}{2}\).

• Observe the known sine values and corresponding angles from the unit circle.
• Recognize symmetries of the sine function, particularly how certain angles yield positive or negative results.
• Appreciate how the domain and range restrictions of inverse trigonometric functions guide the angle selection process.

For the expression\( ext{sin}^{-1}(-\frac{\sqrt{2}}{2})\), we evaluate it by finding the angle \(\theta\) such that the sine of \(\theta\) results in the given value within the accepted range. Here, the correct angle is \(-\frac{\pi}{4}\) or \(-45\) degrees, which aligns with sine's behavior over specified arc regions.