Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that each equation is an identity. $$ \tan \left(\sin ^{-1} x\right)=\frac{x}{\sqrt{1-x^{2}}} $$

Step-by-Step Solution

Answer

The identity is true since \( \tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1-x^2}} \).

Step 1: Understand the Inverse Sine Function

The expression \( \sin^{-1}(x) \) or arcsin \( x \) is the angle \( \theta \) such that \( \sin(\theta) = x \) and \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \). This angle \( \theta \) is needed to evaluate \( \tan(\theta) \).

Step 2: Use the Pythagorean Identity

Because \( \sin(\theta) = x \), the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) implies \( \cos^2(\theta) = 1 - x^2 \). Thus, \( \cos(\theta) = \sqrt{1-x^2} \) because \( \cos(\theta) \) must be non-negative for \( \theta \) in the given range.

Step 3: Solve for Tan of the Angle

Now that you have \( \sin(\theta) = x \) and \( \cos(\theta) = \sqrt{1-x^2} \), you can calculate \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{x}{\sqrt{1-x^2}} \). This confirms that \( \tan(\sin^{-1}(x)) = \frac{x}{\sqrt{1-x^2}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles if we know the value of the trigonometric function of that angle. These functions are the inverse versions of the regular sine, cosine, and tangent functions. For instance, the inverse sine function, denoted as \( \sin^{-1}(x) \) or "arcsin", finds the angle whose sine is \( x \).

Inverse functions like \( \sin^{-1}(x) \) are crucial when solving equations that require knowing an angle instead of its trigonometric ratio. They are widely used in various fields, including physics, engineering, and architecture, to determine angles based on given lengths or distances.

• The range of \( \sin^{-1}(x) \) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), ensuring the result is always an angle where sine values between -1 and 1 can exist.

Understanding inverse trigonometric functions allows us to bridge the gap between angle measures and their trigonometric outputs, making it essential for converting values back into angles in real-world problems.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry, especially helpful in relating the sine and cosine of an angle. It states that \( \sin^2(\theta) + \cos^2(\theta) = 1 \) for any angle \( \theta \). This identity helps in finding unknown trigonometric values when one value is already known.

The identity originates from the Pythagorean Theorem, used in the context of a right triangle. If we consider a right triangle inside a unit circle, the length of the hypotenuse is 1. Hence, the sum of the squares of the other two sides, which correlate with sine and cosine, equals one.

• In this exercise, knowing \( \sin(\theta) = x \) allowed us to utilize the Pythagorean identity to deduce \( \cos^2(\theta) = 1 - x^2 \).
• From the positive range of the arccosine function and the specified range of \( \theta \), we take \( \cos(\theta) = \sqrt{1-x^2} \).

The Pythagorean identity is not just a mathematical curiosity but a powerful tool in solving trigonometric equations and proving various identities.

Tangent Function

The tangent function is another essential component in trigonometry. It is defined as the ratio of the sine and cosine functions: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function is often used to find the slope of a line in coordinate geometry or in calculus to determine the steepness of a curve.

• In the specific solution given, we computed \( \tan(\theta) = \frac{x}{\sqrt{1-x^2}} \) by substituting the known values of sine and cosine.

The tangent function can have values ranging from negative to positive infinity, which results from cosine nearing zero. This behavior signifies that the tangent function becomes undefined precisely at \( \theta = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Understanding the behavior of tangent is crucial, whether analyzing periodic functions or finding angles in triangles when heights and distances are known. Its inverses, often seen as \( \tan^{-1}(x) \), allow for determining angles, much like the other inverse trigonometric functions.