Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(\mathrm{0}<\boldsymbol{\theta}<\pi / 2 .\) Assume \(a>\mathrm{0}.\) $$\sqrt{a^{2}+u^{2}}, \quad u=a \tan \theta$$

Step-by-Step Solution

Answer

The algebraic expression \(\sqrt{2-x^{2}}\) can be written as a trigonometric function of \(\theta\) by substituting \(x = \sqrt{2}\sin\theta\). The final simplified expression is \(\sqrt{2}\cos\theta\).

Step 1: Substitute for \(x\)

Start by substituting \(x = \sqrt{2}\sin\theta\) into the algebraic expression. This gives \(\sqrt{2 - (\sqrt{2}\sin\theta)^{2}}\).

Step 2: Simplify the expression

Next, simplify the expression inside the square root. This becomes \(\sqrt{2 - 2\sin^{2}\theta}\).

Step 3: Recognize a trigonometric identity

The expression under the square root, \(2 - 2\sin^{2}\theta\), can be recognized as a trigonometric identity \(2\cos^{2}\theta\). Substitute this identity into the equation, thus becoming \(\sqrt{2\cos^{2}\theta}\).

Step 4: Simplify the expression further

The square root of \(2\cos^{2}\theta\) is \(\sqrt{2}\cos\theta\), which is our final simplified expression.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations but does not contain an equality sign. In the context of the given exercise, \(\sqrt{2-x^2}\) serves as the algebraic expression. Understanding how to rewrite algebraic expressions in terms of trigonometric functions is a valuable skill. Trigonometric substitution, as used in the exercise, is one method to transform the expression, making it easier to integrate, differentiate, or otherwise manipulate, especially when dealing with integrals or equations that involve square roots.

In the provided exercise, we see a square root of a quadratic expression. Recognizing patterns in these expressions can often hint at useful substitutions that will simplify the problem. The goal with the substitution \(x = \sqrt{2}\sin\theta\) is to utilize the Pythagorean identity of trigonometry to make the under-root expression more manageable.

Trigonometric Function

Trigonometric functions like sine, cosine, and tangent are the fundamental building blocks of trigonometry. They relate the angles of a triangle to the lengths of its sides and are essential in various areas of mathematics, including geometry and calculus. In our exercise, the trigonometric function of interest is the sine function, expressed as \(\sin\theta\).

By replacing \(x\) with \(\sqrt{2}\sin\theta\), we transition from an algebraic expression to one that can be described entirely in terms of trigonometry. Using the substitution not only taps into the power of trigonometric functions to simplify complex algebraic expressions but also leverages their properties to advance the simplification of the expression.

Simplifying Expressions

Simplifying expressions involves reducing a complex expression to its simplest form, making it easier to interpret or solve. In trigonometric substitution problems like ours, simplification often relies on recognizing patterns that match trigonometric identities. After substituting \(x = \sqrt{2}\sin\theta\), our next step is to simplify \(2 - (\sqrt{2}\sin\theta)^2\).

This simplification involves basic algebraic manipulation such as squaring a binomial and combining like terms. The challenge is to not only perform the arithmetic but also to look for opportunities where a trigonometric identity can be used to replace an algebraic expression, thus further simplifying the equation. The ability to simplify complex expressions is a testament to the synergy between algebra and trigonometry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. These identities are immensely useful in simplifying trigonometric expressions and solving equations. One of the most fundamental identities is the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), which is used in our exercise.

In Step 3, by noticing that \(2 - 2\sin^2\theta\) resembles the rearranged form of the Pythagorean identity \(2\cos^2\theta\), we can substitute directly, transforming the expression into a simpler form. Recognizing and applying trigonometric identities correctly is crucial in solving a wide range of mathematical problems, including those involving trigonometric substitution, as it simplifies the problem to a more manageable state. Mastery of these identities is essential for students to navigate through trigonometry efficiently.