Problem 10 - Precalculus with Calculus Previews

Question

Find the exact value of the given trigonometric expression. Do not use a calculator. $$ \sin ^{-1} \frac{\sqrt{2}}{2} $$

Step-by-Step Solution

Answer

\(\sin^{-1}\frac{\sqrt{2}}{2} = \frac{\pi}{4}\).

Step 1: Understand the Problem

We need to find the angle whose sine value is \( \frac{\sqrt{2}}{2} \), denoted as \( \sin^{-1} \frac{\sqrt{2}}{2} \). This is an inverse trigonometric function.

Step 2: Recall the Range of Arcsine Function

The arcsine function, \(\sin^{-1}(x)\), provides an angle from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). We need our solution to fit within this interval.

Step 3: Identify the Special Angle

The value \( \frac{\sqrt{2}}{2} \) corresponds to one of the special angles in trigonometry. Knowing that \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), the angle satisfying \( \sin \theta = \frac{\sqrt{2}}{2} \) within the desired range is \( \frac{\pi}{4} \).

Step 4: Validate the Solution

Verify that \( \frac{\pi}{4} \) is within the range \(-\frac{\pi}{2}, \frac{\pi}{2}\). Since \( \frac{\pi}{4} \) is \(45^\circ\) and falls in the first quadrant, it is within the valid range for arcsine.

Key Concepts

Arcsine Function

The arcsine function, denoted as \( \sin^{-1}(x) \), is an inverse trigonometric function. It is used to determine an angle when the sine value is known. Unlike the sine function, which can have an output anywhere between -1 and 1, the arcsine function returns an angle, restricted to a specific interval. This interval is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), or from -90° to 90°. Only angles within this range are valid for the arcsine function.
This function helps us solve equations like \( \sin \theta = x \) when we need to know \( \theta \). For example, if \( \sin \theta = \frac{\sqrt{2}}{2} \), we find \( \theta \) by determining which angle has a sine of \( \frac{\sqrt{2}}{2} \) inside the \(-\frac{\pi}{2} \text{ to } \frac{\pi}{2}\) range.
Understanding the output restrictions of arcsine is crucial:

• The arcsine of a number greater than 1 or less than -1 is undefined as no sine value exists outside this range.
• The arcsine function is most useful when dealing with real-world scenarios that require angle determination from known ratios.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved angles. These identities are essential tools in trigonometry as they simplify complex trigonometric expressions and solve equations effectively.
Some key trigonometric identities involve functions like sine, cosine, and tangent:

• The Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Angle Addition Formulas: \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \)
• Double Angle Formulas: \( \sin(2\theta) = 2\sin\theta \cos\theta \)

Knowing the arcsine function’s specific value provides insight into other identities. For instance, if \( \theta = \frac{\pi}{4} \) leads to \( \sin \theta = \frac{\sqrt{2}}{2} \), we can apply this to calculate or derive values using identities like those mentioned above.
Mastering trigonometric identities not only aids solving trigonometric equations but also enhances understanding of inverse functions and special angle calculations.

Special Angles

Trigonometry often deals with specific angles known as "special angles." These include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\) and their corresponding degrees \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}\). Each of these angles has well-known sine, cosine, and tangent values that form the foundation for solving trigonometric problems.
For example:

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

In the arcsine context, identifying these special angles is vital as they are often used to determine the exact value of an expression. For \( \sin^{-1}(\frac{\sqrt{2}}{2}) \), recognizing that \( \frac{\pi}{4} \) is a special angle where the sine equals \( \frac{\sqrt{2}}{2} \) quickly provides the solution.
Familiarity with these special angles and their trigonometric values allows ease in solving numerous trigonometric expressions without needing a calculator, making calculations faster and more intuitive.