Problem 42 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in \(x .\) \(\sin \left(\tan ^{-1} x\right)\)

Step-by-Step Solution

Answer

\( \sin(\tan^{-1} x) = \frac{x}{\sqrt{x^2 + 1}} \).

Step 1: Understand the Problem

We need to rewrite the trigonometric expression \( \sin(\tan^{-1} x) \) in terms of \( x \). This involves expressing the sine of an angle whose tangent is \( x \).

Step 2: Use the Right Triangle Definition

Interpret \( \tan^{-1} x = \theta \) to mean that \( \tan \theta = x \). In a right triangle, this corresponds to \( \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{1} \). Here, suppose the opposite side is \( x \) and the adjacent side is \( 1 \).

Step 3: Determine the Hypotenuse

Apply the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \].

Step 4: Compute \( \sin \theta \)

Using the triangle, we calculate \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{\sqrt{x^2 + 1}} \).

Step 5: Simplify the Expression

The expression \( \sin(\tan^{-1} x) \) is therefore rewritten as:\[ \frac{x}{\sqrt{x^2 + 1}} \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when we know the sides of a triangle. For example, when you see \( \tan^{-1} x \), this means you are looking for an angle whose tangent is \( x \). This function, also known as the arctangent, is important in solving problems that involve angles of right triangles.
When solving problems with inverse trigonometric functions, it's useful to think about what an angle represents.

• \( \tan^{-1} x \) provides the angle whose tangent is \( x \).
• It helps convert a ratio of sides back into an angle.

Using these functions is like having a key to switch between geometric relationships and algebraic expressions. This makes them invaluable for calculations involving trigonometry, where one might not have the angle readily available.

Right Triangle Trigonometry

Right triangle trigonometry is all about understanding relationships between the angles and sides of a right triangle. For instance, if you know the tangent of an angle, you can use it to understand how the opposite side relates to the adjacent side of the triangle.
When dealing with \( \tan \theta = x \), it translates into a right triangle:

• The opposite side corresponds to \( x \).
• The adjacent side corresponds to 1.

To make this visible:

• Consider the use of a simple triangle with these side lengths.
• You can visualize how the sides form the triangle, providing a clearer pathway to using trigonometric identities.

This foundational concept behind right triangle trigonometry makes it much easier to transition between geometric shapes and algebraic equations.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry which describes the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Using the triangle from our previous explanation where:

• The opposite side is \( x \).
• The adjacent side is 1.

We apply the Pythagorean theorem:

• Calculate the hypotenuse as \( \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).

This relationship not only finds the length of the unknown side but simplifies trigonometric problems by introducing the concept of ratios. It allows for a systematic approach in transitioning expressions, like converting \( \sin(\tan^{-1} x) \) into a simpler algebraic form \( \frac{x}{\sqrt{x^2 + 1}} \).