Problem 133 - Precalculus with Limits

Question

In Exercises 131 - 134, write the trigonometric expression as an algebraic expression. \( \cos\left(2 \arcsin x\right) \)

Step-by-Step Solution

Answer

\(\cos(2 \arcsin x) = 1 - 2x^2\)

Step 1: Setup

Let's consider the double-angle identity in the form of cosine, which is \( \cos(2a) = 1 - 2 \sin^2(a) \). Here, \( a \) is the arcsin of \( x \), or \( a = \arcsin x \), which means \( \sin a = x \). Now, we substitute \( a \) for \( \arcsin x \) in the identity.

Step 2: Substitute

Now, if we replace `a` in our identity with \(\arcsin x\), we get \(\cos(2 \arcsin x) = 1 - 2 \sin^2(\arcsin x)\). The term \(\sin(\arcsin x)\) can be simplified to \(x\), so we plug this into the equation.

Step 3: Simplify

By substituting \(x\) for \(\sin(\arcsin x)\), we finally get \(\cos(2 \arcsin x) = 1 - 2x^2\). This is the final step to convert the given trigonometric expression into an algebraic expression.

Key Concepts

Double-Angle Identities

Double-angle identities are powerful tools in trigonometry. They allow us to express trigonometric functions involving double angles in terms of single angles. This is particularly useful when simplifying complex expressions.

One of the commonly used double-angle identities is for cosine, given by:

• \( \cos(2a) = 1 - 2 \sin^2(a) \)

This identity enables you to break down cosine of double angles into simpler terms involving sine. When solving problems, knowing when to apply these identities can make finding solutions much quicker and more intuitive.

The identity shows that by using a known angle 'a', you can calculate the cosine of its double '2a' through its corresponding sine value. This process significantly simplifies tasks in trigonometry where calculating angles of twice the magnitude is required.

Algebraic Expressions

Algebraic expressions are a way to write numbers and variables in a formulaic manner to solve problems algebraically. They often involve variables, constants, and arithmetic operations.

Converting trigonometric expressions into algebraic forms can often simplify the evaluation of these expressions.

For example, in the expression \( \cos(2 \arcsin x) \), we use a double-angle identity to rewrite it as an algebraic expression. Once the identity \( \cos(2a) = 1 - 2 \sin^2(a) \) is applied and \( a \) is substituted with \( \arcsin x \), we simplify it further as:

• \( \cos(2 \arcsin x) = 1 - 2x^2 \)

This transformation takes a trigonometric concept and changes it into a format that might be easier to work with, especially in algebraic equations.

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the effect of the original trigonometric functions. They allow us to find angles when given trigonometric ratios.

Functions like arcsine, arccosine, and arctangent play a crucial role in finding angles from specific values in trigonometry.

• \( \arcsin(x) \), for example, finds an angle whose sine is \( x \).

In our exercise, \( \arcsin x \) helps us identify angle 'a', where \( \sin a = x \).

This angle is then plugged into trigonometric identities, such as the double-angle identity, to facilitate the conversion of trigonometric expressions into algebraic ones. Inverse trigonometric functions are indispensable in calculus and trigonometry for solving equations and evaluating expressions involving angles.