Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the exact value without using a calculator. $$ \sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

The exact value is \(-\frac{\pi}{4}\).

Step 1: Understand the Problem

We need to find the angle whose sine value is \(-\frac{\sqrt{2}}{2}\), and this angle will be in the range of the inverse sine function, \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 2: Recall Special Angles

Remember that \(\sin\theta = -\frac{\sqrt{2}}{2}\) is a familiar value. The value \(\frac{\sqrt{2}}{2}\) is associated with an angle of \(\frac{\pi}{4}\) or \(-\frac{\pi}{4}\), where the sine value can be negative.

Step 3: Determine the Specific Angle

Within the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (negative to positive angles), the angle that satisfies \(\sin \theta = -\frac{\sqrt{2}}{2}\) is \(-\frac{\pi}{4}\). This angle corresponds to a negative value in the first quadrant when considering the unit circle.

Step 4: Conclude the Angle

The exact value of \(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) is \(-\frac{\pi}{4}\) as it fits within the defined range of the inverse sine function.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find angles when given values of trigonometric ratios. They are essentially the reverse operations of the standard trigonometric functions like sine, cosine, and tangent. For example, the inverse sine function (written as \( \sin^{-1} \) or arcsin) determines the angle that has a given sine value.

The range of the inverse sine function is very specific:

• It is limited to producing angles between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• This limitation is set to ensure that the function is a true inverse, as sine is a many-to-one function in its natural range.
• Thus, using this limited range allows each sine value to map to only one angle.

For example, finding \( \sin^{-1}(-\frac{\sqrt{2}}{2}) \) means determining which angle within this range has a sine value of \(-\frac{\sqrt{2}}{2}\).

Special Angles

Special angles are specific angles that have well-known trigonometric values, often found in problems to avoid the need for a calculator. Common special angles include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2}\), along with their negatives and complements.

For the sine function, the special angle \(\frac{\pi}{4}\) (or 45 degrees) is particularly important because:

• Its sine value is \(\frac{\sqrt{2}}{2}\).
• The negative of this value is \(-\frac{\sqrt{2}}{2}\), occurring at \(-\frac{\pi}{4}\) or 315 degrees when considering the standard Cartesian plane.

These angles help us quickly identify solutions to trigonometric equations without a calculator, utilizing basic geometric understanding and often built-in memorization of these key values.

Sine Function

The sine function relates an angle in a right-angled triangle to the ratio of the length of the side opposite the angle to the hypotenuse. It is a periodic function, leading it to repeat values over a set interval, typically \(2\pi\) or 360 degrees.

Some key features include:

• The sine function oscillates between \(-1\) and \(1\).
• It's positive in the first and second quadrants, while negative in the third and fourth.
• Because the sine value repeats, within a single cycle, there may be two angles sharing the same sine value, thus necessitating the use of inverse functions to determine which angle is desired.

This periodic nature and symmetry around the origin are essential factors when solving equations using the sine function.

Unit Circle

The unit circle is a fundamental concept in trigonometry, serving as a tool to easily visualize trigonometric functions and their relationships. It is a circle with a radius of 1 and is centered at the origin of the coordinate plane.

Here's why it's useful:

• The angle \(\theta\) in radians is measured from the positive x-axis, creating specific coordinates where the circle intersects \(\cos(\theta)\) and \(\sin(\theta)\) represent the x and y coordinates, respectively.
• This visual tool is invaluable for understanding how the sine function behaves for different angles, especially with negative ones found in the inverse trigonometric problems.
• When considering \(\sin^{-1}(-\frac{\sqrt{2}}{2})\), the unit circle helps identify that the corresponding angle \(-\frac{\pi}{4}\) reflects the sine value's symmetry across the origin.

The unit circle thus simplifies finding and understanding trigonometric values, turning abstract representations into intuitive concepts.