Problem 35 - Precalculus with Calculus Previews

Question

Write the given expression as an algebraic expression in \(x\). $$ \tan (\arcsin x) $$

Step-by-Step Solution

Answer

\( \tan(\arcsin x) = \frac{x}{\sqrt{1-x^2}} \)

Step 1: Understanding the Problem

We begin with the trigonometric expression \( \tan(\arcsin x) \). This asks us to find the tangent of the angle whose sine is \( x \). We'll use trigonometric identities to express this in terms of \( x \).

Step 2: Draw a Right Triangle

To visualize this problem, consider a right triangle where one of the angles is \( \arcsin x \). This means the opposite side is \( x \), and the hypotenuse is 1 (since sine is opposite over hypotenuse).

Step 3: Calculate the Adjacent Side

Using the Pythagorean theorem, we find the adjacent side of the triangle. Since we have the hypotenuse and the opposite, the formula is \( \text{adjacent} = \sqrt{1^2 - x^2} = \sqrt{1-x^2} \).

Step 4: Express Tangent in Terms of Triangle Sides

The tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, \( \tan(\arcsin x) = \frac{x}{\sqrt{1-x^2}} \).

Step 5: Simplify the Expression

We've found \( \tan(\arcsin x) \) in terms of \( x \): \( \frac{x}{\sqrt{1-x^2}} \). This is the algebraic expression we're looking for.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us determine angles given a ratio of sides in a right triangle. They are the opposite operations of the standard trigonometric functions, such as sine, cosine, and tangent. When you see something like \( \arcsin \) or \( \arccos \), this means we are trying to find the angle whose sine or cosine equals a given number.
For example, \( \arcsin(x) \) is the angle whose sine value is \( x \). It is important to remember that the range for \( \arcsin \) is limited to \([-\frac{\pi}{2}, \frac{\pi}{2}]\), ensuring the function returns a unique angle. This concept is essential for converting trigonometric expressions into algebraic expressions involving angles.

Right Triangle

Right triangles are a fundamental concept in trigonometry. They include one angle that is exactly 90 degrees, known as a right angle. The side opposite the right angle is called the hypotenuse, and it is always the longest side of the triangle.
The two other sides are referred to as the opposite and adjacent sides, relative to a given angle. The opposite side is the one directly across from the angle in question, and the adjacent side is next to the angle. Understanding these sides and their relationships is key when using trigonometric functions and solving problems like the original exercise.

Pythagorean Theorem

The Pythagorean theorem is a widely-used principle in geometry, specifically with right triangles. It states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as \( c^2 = a^2 + b^2 \) where \( c \) is the hypotenuse, \( a \) is the opposite side, and \( b \) is the adjacent side.
This theorem allows us to determine one side of the triangle if the other two are known. In the original problem, we found the adjacent side using the hypotenuse and the opposite side with the formula \( \text{adjacent} = \sqrt{1-x^2} \). This step is crucial to finding expressions involving trigonometric identities and applying them to inverse functions.