Problem 46 - Algebra and Trigonometry

Question

\(43-46\). Write the given expression as an algebraic expression in \(x\). $$ \cos \left(2 \sin ^{-1} x\right) $$

Step-by-Step Solution

Answer

The expression \( \cos(2\sin^{-1} x) \) simplifies to \( 1 - 2x^2 \).

Step 1: Understand the Inverse Function

The expression \( \cos(2 \sin^{-1} x) \) involves an inverse trigonometric function. The \( \sin^{-1} x \) or arcsin function results in an angle \( \theta \) such that \( \sin \theta = x \). Here, \( \theta = \sin^{-1} x \).

Step 2: Use a Trigonometric Identity

To simplify \( \cos(2\theta) \), we can use the double angle identity for cosine: \( \cos(2\theta) = 1 - 2\sin^2\theta \).Since \( \theta = \sin^{-1} x \), we have \( \sin \theta = x \), thus \( \sin^2 \theta = x^2 \).

Step 3: Substitute Trigonometric Identity

Substitute \( \sin^2 \theta = x^2 \) into the identity:\[ \cos(2\theta) = 1 - 2\sin^2\theta = 1 - 2x^2 \]

Step 4: Write the Expression

Hence, we have rewritten \( \cos(2\sin^{-1} x) \) as the algebraic expression \( 1 - 2x^2 \).

Key Concepts

trigonometric identities

Trigonometric identities are essential tools in simplifying and transforming trigonometric expressions. They provide relationships between the trigonometric functions, which can be used to simplify expressions, solve equations, and evaluate integrals, among other things.
One popular set of trigonometric identities involves the basic trigonometric functions like sine, cosine, and tangent. These identities take advantage of the relationships between the sides and angles in a right triangle.
Some commonly used identities include:

• Pythagorean identity: \( an^2 \theta + 1 = \sec^2 \theta\)
• Reciprocal identities: \(\csc \theta = \frac{1}{\sin \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\)
• Quotient identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

By understanding and applying these identities, we can manipulate and simplify expressions to achieve desired forms, like turning inverse trigonometric functions involving angles into algebraic expressions.

double angle identity

The double angle identity is a significant and useful identity in trigonometry that allows us to express functions involving "doubled" angles in terms of single angles. This identity can simplify problems that involve more complex trigonometric functions. For cosine, the double angle identity is particularly handy.
The double angle identity for cosine states that:\[\cos(2\theta) = \cos^2 \theta - \sin^2 \theta .\] This can also be rewritten using other common trigonometric identities, providing alternative forms like:

• \(\cos(2\theta) = 1 - 2\sin^2 \theta\)
• \(\cos(2\theta) = 2\cos^2 \theta - 1\)

In the context of the problem, we used the alternative form \(\cos(2\theta) = 1 - 2\sin^2 \theta\). By understanding how sine and cosine relate to each other, we substituted \(\sin^2 \theta\) with \(x^2\), transforming the trigonometric expression into an algebraic one. This shows the power and versatility of the double angle identity in simplifying expressions and solving trigonometric problems.

algebraic expressions

Algebraic expressions involve numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. The goal of translating trigonometric expressions into algebraic ones is to express them in their simplest possible form without trigonometric function references.
Starting with trigonometric identities, such as those involving inverse trigonometric functions, let's consider turning them into algebraic expressions:

• The expression \(\cos(2\sin^{-1} x)\) is simplified using the double angle identity.
• Knowing \(\sin^{-1} x\) yields an angle \(\theta\) such that \(\sin \theta = x\), we substitute \(\sin^2 \theta = x^2\).
• Finally, the expression becomes an algebraic form: \(1 - 2x^2\).

Simplifying trigonometric expressions into algebraic ones makes them more manageable for various applications. With the use of algebraic expressions, students can integrate them into different mathematical problems, making both computation and understanding easier.