Problem 3 - Precalculus

Question

Find the exact value. \(\arcsin \left(-\frac{\sqrt{2}}{2}\right)\)

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 \(-\frac{\sqrt{2}}{2}\). This is essentially the definition of the arcsin function, which returns the angle whose sine is the given value.

Step 2: Identify the Range for Arcsin

The inverse sine function, \(\arcsin(x)\), returns angles in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (or \([-90^\circ, 90^\circ]\)). Therefore, the answer must lie within this range.

Step 3: Recognize Known Values of Sine

Recall that \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\). This means \(-\frac{\pi}{4}\) radians (or \(-45^\circ\)) is an angle in the desired range with the correct sine value.

Step 4: Verify the Angle

Confirm that \(-\frac{\pi}{4}\) is indeed within the specified range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), which it is. Thus, the correct value for \(\arcsin\left(-\frac{\sqrt{2}}{2}\right)\) is \(-\frac{\pi}{4}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics. They help us find angles when we know the value of the sine, cosine, or other trigonometric ratios.
In the world of trigonometry, these inverse functions reverse the original trigonometric functions. This is key when solving equations involving angles and sides of right triangles.
For example:

• The inverse of the sine function is called arcsin, often denoted as \(\arcsin(x)\).
• The inverse of the cosine function is arccos, denoted by \(\arccos(x)\).
• Similarly, arctan is the inverse of the tangent function, written as \(\arctan(x)\).

These functions allow us to determine the angle associated with a given trigonometric ratio. This is exactly what we are doing when solving for \(\arcsin\left(-\frac{\sqrt{2}}{2}\right)\). We are finding the angle whose sine value is \(-\frac{\sqrt{2}}{2}\).
These inverse functions are restricted to specific ranges to ensure they behave consistently and give unique outputs, which is crucial for solving mathematical problems.

Angle Measurement

Angles can be measured in different units, with degrees and radians being the most common. Understanding these measurement systems is crucial when working with trigonometric functions.
Degrees are more familiar to most people:

• A full circle is 360 degrees.
• 90 degrees is a right angle.
• 45 degrees is half of a right angle.

Radians, however, are more commonly used in mathematics due to their direct relationship with the unit circle. Here’s a quick conversion to remember:

• A full circle is \(2\pi\) radians.
• 90 degrees equals \(\frac{\pi}{2}\) radians.
• 45 degrees equals \(\frac{\pi}{4}\) radians.

When dealing with arcsin values like \(-\frac{\pi}{4}\), it's important to know these conversions. It helps in understanding whether the solution involves a positive or negative angle, and fits the expected range of the inverse trig function.

Sine Values

Sine values represent the y-coordinate of a point on the unit circle. They range between -1 and 1, reflecting the maximum stretch of the unit circle vertically.
Some sine values are easily recognizable and very significant in trigonometry:

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

In this exercise, the sine value \(-\frac{\sqrt{2}}{2}\) corresponds to the angles \(-\frac{\pi}{4}\) and \(\frac{3\pi}{4}\), but only the angle that lies within the arcsin range is valid for our purpose. Understanding these values can simplify finding the angle that corresponds to a given sine value.

Unit Circle

The unit circle is a vital concept in trigonometry that greatly simplifies the understanding of angles and their sine values. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
Each point on the unit circle corresponds to

• A specific angle measure from the positive x-axis.
• A set of coordinates \((x, y)\). The x-coordinate represents the cosine, and the y-coordinate represents the sine of the angle.

The unit circle is immensely helpful for visualizing the sine value of an angle. For example:

• At \(45^\circ\) or \(\frac{\pi}{4}\) radians, the coordinates are \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\).
• Its reflection in the third quadrant, \(-\frac{\pi}{4}\), gives the coordinates \(\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)\).
• This helps us pinpoint the angle with sine value \(-\frac{\sqrt{2}}{2}\) while considering the range of the inverse sine function.

Thus, the unit circle not only provides a geometric representation of trigonometric functions but also aids in understanding the reasoning behind inverse trigonometric function ranges.