Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the trigonometric expression as an algebraic expression. $$\cos (\arcsin x-\arctan 2 x)$$

Step-by-Step Solution

Answer

Thus, the given trigonometric expression simplifies to \(\frac{\sqrt{1 - x^2} + 2x^2}{\sqrt{1 + 4x^2}} \).

Step 1: Apply cosine difference identity

First apply the cosine difference identity, which states that \( \cos(A - B) = cos(A)cos(B) + sin(A)sin(B) \). Using it on our problem gives us \( \cos(\arcsin x - \arctan 2x) = \cos(\arcsin x)\cos(\arctan 2x) + \sin(\arcsin x)\sin(\arctan 2x) \).

Step 2: Use properties of inverse trigonometric functions

Next, recall the identities: \(\cos(\arcsin x) = \sqrt{1 - x^2} \), \(\sin(\arcsin x) = x\), \(\cos(\arctan x) = \frac{1}{\sqrt{1 + x^2}}\), and \( \sin(\arctan x) = \frac{x}{\sqrt{1 + x^2}} \). Substituting these in the simplified expression leads to \( \sqrt{1 - x^2} \cdot \frac{1}{\sqrt{1 + 4x^2}} + x \cdot \frac{2x}{\sqrt{1 + 4x^2}} \)

Step 3: Simplify Resulting Expression

Simplifying the above expression gives us \( \frac{\sqrt{1 - x^2}}{\sqrt{1 + 4x^2}} + \frac{2x^2}{\sqrt{1 + 4x^2}} \). These two fractions have the same denominator, so they can be combined. The final algebraic expression is \(\frac{\sqrt{1 - x^2} + 2x^2}{\sqrt{1 + 4x^2}} \)

Key Concepts

Cosine Difference Identity

The cosine difference identity is a powerful tool in trigonometry that helps to simplify expressions involving the difference of two angles. It states that for any two angles \(A\) and \(B\), the cosine of their difference is given by the formula:

• \( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \)

This identity is useful because it allows us to break a complex expression into simpler components. In the given exercise, this identity helps to transform the trigonometric expression \( \cos(\arcsin x - \arctan 2x) \) into a sum of products of sines and cosines of individual angles.
First, recognize \( \arcsin x \) and \( \arctan 2x \) as the two angles whose difference we are considering. By applying the identity, we rewrite the expression as:

• \( \cos(\arcsin x)\cos(\arctan 2x) + \sin(\arcsin x)\sin(\arctan 2x) \)

This sets the stage for further simplification using the properties of inverse trigonometric functions.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine an angle given a trigonometric ratio. Each function maps a ratio back to an angle. Some important properties for the common inverse trigonometric functions include:

• \( \cos(\arcsin x) = \sqrt{1 - x^2} \)
• \( \sin(\arcsin x) = x \)
• \( \cos(\arctan x) = \frac{1}{\sqrt{1 + x^2}} \)
• \( \sin(\arctan x) = \frac{x}{\sqrt{1 + x^2}} \)

These properties arise from the definitions of \( \arcsin x \) and \( \arctan x \) on the unit circle and right triangle relationships.
In our expression \( \cos(\arcsin x - \arctan 2x) \), applying these properties simplifies the cosine and sine terms individually:

• \( \cos(\arcsin x)\) becomes \( \sqrt{1 - x^2} \)
• \( \cos(\arctan 2x)\) simplifies to \( \frac{1}{\sqrt{1 + 4x^2}} \)
• \( \sin(\arcsin x)\) remains as \( x \)
• \( \sin(\arctan 2x)\) becomes \( \frac{2x}{\sqrt{1 + 4x^2}} \)

By substituting these simplified forms back into the cosine difference identity, the trigonometric expression starts to resemble an algebraic expression.

Algebraic Simplification

Algebraic simplification is the process of making a mathematical expression as straightforward and readable as possible. In this exercise, after applying trigonometric identities, our task is to simplify the resulting algebraic expressions.
We begin with the expression obtained in the previous step:

• \( \sqrt{1 - x^2} \cdot \frac{1}{\sqrt{1 + 4x^2}} + x \cdot \frac{2x}{\sqrt{1 + 4x^2}} \)

Notice that both terms share the same denominator \( \sqrt{1 + 4x^2} \). This allows us to combine them into a single fraction:

• \( \frac{\sqrt{1 - x^2} + 2x^2}{\sqrt{1 + 4x^2}} \)

Combining fractions via a common denominator is a core technique in algebra. It simplifies the expression and makes it easier to interpret or use in further calculations. In the final expression, each component is expressed in its simplest form, thereby achieving the goal of turning a trigonometric expression into a clear algebraic one.