Problem 3 - Essential Calculus Early Transcendentals

Question

Find the exact value of each expression. (a) arctan 1 (b) \(\sin ^{-1}(1 / \sqrt{2})\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{4}\); (b) \(\frac{\pi}{4}\).

Step 1: Understanding Inverse Functions

The problem requires us to find the exact values of inverse trigonometric functions: arctan and arcsine. These functions give the angle whose tangent or sine is a given value.

Step 2: Finding arctan(1)

The function \( ext{arctan}(x)\) returns the angle whose tangent is \(x\). We are looking for the angle \(\theta\) such that \(\tan(\theta) = 1\). Since \(\tan(\frac{\pi}{4}) = 1\), the exact value is \(\frac{\pi}{4}\).

Step 3: Finding \(\sin^{-1}(\frac{1}{\sqrt{2}})\)

The function \(\sin^{-1}(x)\) or \(\arcsin(x)\) returns the angle whose sine is \(x\). We want \(\theta\) such that \(\sin(\theta) = \frac{1}{\sqrt{2}}\). Since \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\), the exact value is \(\frac{\pi}{4}\).

Key Concepts

arctan function

The arctan function, also known as the inverse tangent function, is an essential concept in trigonometry. It helps us find the angle whose tangent is a given number. For example, when you encounter a problem asking for \( \text{arctan}(1) \), you need to understand it as the angle \( \theta \) such that \( \tan(\theta) = 1 \). The result is an angle in radians, typically ranging between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). This is because the arctan function is defined within this interval to ensure it outputs a unique angle for each input.

• The tangent of an angle is simply the ratio of the opposite side to the adjacent side in a right triangle.
• To solve \( \text{arctan}(1) \), recall that \( \tan\left(\frac{\pi}{4}\right) = 1 \), so \( \text{arctan}(1) = \frac{\pi}{4} \).

This makes \( \frac{\pi}{4} \) the exact value we are looking for when we apply the arctan function to 1.

arcsine function

The arcsine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), determines the angle whose sine is \( x \). It's a fundamental inverse trigonometric function utilized to find angles from known sine values. Typically, the output of an arcsine function is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), ensuring each sine value has a unique corresponding angle. When dealing with problems like \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \), you're looking for the angle \( \theta \) such that \( \sin(\theta) = \frac{1}{\sqrt{2}} \).

• This sine value is commonly associated with \( \frac{\pi}{4} \), a special angle where both sine and cosine equal \( \frac{1}{\sqrt{2}} \).
• Thus, the exact value for \( \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \) is \( \frac{\pi}{4} \).

Recognizing these special angles, such as \( \frac{\pi}{4} \), \( \frac{\pi}{6} \), and \( \frac{\pi}{3} \), is crucial for quickly solving trigonometric problems.

exact values of trigonometric expressions

Exact values of trigonometric expressions are critical in various math applications. These are values expressed precisely, rather than using decimal approximations. Recognizing these values helps simplify solving trigonometric equations efficiently. Some frequently encountered exact values arise from key angles like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \frac{\pi}{2} \).

• At \( \frac{\pi}{4} \), both sine and cosine equal \( \frac{1}{\sqrt{2}} \), making it a significant angle in trigonometry.
• The tan of \( \frac{\pi}{4} \) is 1, which is a handy fact, especially for the arctan function.
• Memorizing these key angle values helps in deriving solutions effectively.

Understanding these basic trigonometric function values and their connected angles boosts your problem-solving speed and accuracy, making it easier to handle more complex trigonometric challenges.