Problem 3 - Algebra and Trigonometry

Question

Find the exact value of each expression, if it is defined. (a) \(\sin ^{-1} \frac{\sqrt{2}}{2}\) (b) \(\cos ^{-1} \frac{\sqrt{3}}{2}\) (c) \(\sin ^{-1}\left(-\frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

(a) \( \frac{\pi}{4} \), (b) \( \frac{\pi}{6} \), (c) \(-\frac{\pi}{4}\).

Step 1: Understand the Problem

We need to find the exact values of the inverse trigonometric functions. These values tell us the angles for which the trigonometric functions have the given values. We need to identify the angles in radians because it's standard to use radians for these calculations.

Step 2: Solve Part (a)

To find \( \sin^{-1} \frac{\sqrt{2}}{2} \), we need the angle whose sine is \( \frac{\sqrt{2}}{2} \). In the unit circle, this occurs at \( \frac{\pi}{4} \) (also known as 45 degrees) in the first quadrant, where the range for \( \sin^{-1} \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). So the result is \( \frac{\pi}{4} \).

Step 3: Solve Part (b)

For \( \cos^{-1} \frac{\sqrt{3}}{2} \), we need the angle whose cosine is \( \frac{\sqrt{3}}{2} \). On the unit circle, this angle is \( \frac{\pi}{6} \) (30 degrees), where the range for the cosine inverse function is \([0, \pi]\). Therefore, the exact value is \( \frac{\pi}{6} \).

Step 4: Solve Part (c)

With \( \sin^{-1}(-\frac{\sqrt{2}}{2}) \), we need the angle with a sine of \(-\frac{\sqrt{2}}{2}\). On the unit circle, this occurs at \(-\frac{\pi}{4}\), which is within the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) for the inverse sine function. Thus, the exact value is \(-\frac{\pi}{4}\).

Key Concepts

Unit Circle

The Unit Circle is a powerful tool used in trigonometry to define the sine, cosine, and tangent functions. Think of it as a circle with a radius of 1, centered at the origin of a coordinate system.
Here are key points about the Unit Circle:

• Every angle in the unit circle can be represented as a point \( (x, y) \) where \( x \) corresponds to the cosine and \( y \) to the sine of that angle.
• The circle is divided into quadrants. The first quadrant covers angles from \( 0 \) to \( \frac{\pi}{2}\). The second quadrant covers angles from \( \frac{\pi}{2} \) to \( \pi \), the third from \( \pi \) to \( \frac{3\pi}{2}\), and the fourth covers from \( \frac{3\pi}{2} \) to \( 2\pi \).
• Understanding the position of angles on the unit circle is essential for finding the exact trigonometric values, especially when dealing with inverse functions.

The unit circle allows us to visually identify specific angle measures, which makes it easier to determine the exact trigonometric values needed in exercises like finding \( \sin^{-1} \) or \( \cos^{-1} \) values.

Radian Measure

Radian Measure is another way to express the size of an angle, aside from degrees. A complete circle is \( 2\pi \) radians.
Here are some basics about radians:

• The radian is based on the radius of the circle. One radian is the angle formed when the arc length is equal to the radius.
• Common angles in radians include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), and \( \frac{\pi}{2} \).
• Radians are preferred in mathematics because they simplify the calculation of derivatives and integrals in calculus.

In exercises that seek exact trigonometric values, always express your answers in radians. It's standard practice and aligns with how angles are most commonly represented in higher mathematics.

Exact Trigonometric Values

Exact Trigonometric Values are specific values for trigonometric functions that correspond to standard angles.
These values can be determined using the unit circle and known angle measures:

• For \( \sin\) and \( \cos\), common values are \( 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \) and \(1\).
• Each of these values corresponds to specific angles. For example, \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\) and \( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\).
• These are often memorized, as they frequently appear in problems involving trigonometric functions and their inverses.

By memorizing these exact values, you simplify the process of solving problems involving inverse trigonometric functions, as demonstrated by matching given values to their corresponding angles in both the positive and negative ranges.