Question

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ Verify relation \(A\) by differentiating \(x=2 \tan ^{-1} u\). Verify relations \(B\) and \(C\) using a right-triangle diagram and the double-angle formulas $$\sin x=2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right) \text { and } \cos x=2 \cos ^{2}\left(\frac{x}{2}\right)-1$$

Short answer

Question: Verify the following relations when using the substitution \(u = \tan(x/2)\) or \(x = 2\tan^{-1}(u)\) to convert an integrand with trigonometric functions to a rational integrand: A) \(dx = \frac{2}{1+u^2}du\) B) \(\sin x = \frac{2u}{1+u^2}\) C) \(\cos x = \frac{1-u^2}{1+u^2}\)

Step-by-step solution

Verify relation A

To verify relation A, we will find the derivative of \(x = 2\tan^{-1}(u)\) with respect to \(u\). We have: $$\frac{dx}{du} = 2\frac{d\,\tan^{-1}(u)}{du}$$ Now, we recall that the derivative of \(\tan^{-1}(u)\) is \(\frac{1}{1+u^2}\). So we have: $$\frac{dx}{du} = 2\cdot\frac{1}{1+u^2}$$ And therefore, $$dx = \frac{2}{1+u^2}du$$ which verifies relation A.

Right-Triangle Diagram

To verify relations B and C, we will create a right-triangle diagram using the definition of \(u = \tan(x/2)\). We start by constructing a right-angled triangle with one of the angles as \(x/2\), and use the definition of \(\tan(x/2) = \frac{\text{opposite}}{\text{adjacent}}\). The length of the opposite side will be \(u\), adjacent side will be \(1\), and the hypotenuse using Pythagoras theorem is \(\sqrt{1+u^2}\). This triangle will be used to derive expressions for other trigonometric functions.

Verify relation B

To verify relation B, we recall the double-angle formula for \(\sin(x)\): $$\sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$ Using our right-triangle, we determine \(\sin(x/2)\) and \(\cos(x/2)\): $$\sin\left(\frac{x}{2}\right) = \frac{u}{\sqrt{1+u^2}}$$ $$\cos\left(\frac{x}{2}\right) = \frac{1}{\sqrt{1+u^2}}$$ Now, we substitute these expressions into the double-angle formula for \(\sin(x)\): $$\sin x = 2\cdot \frac{u}{\sqrt{1+u^2}}\cdot\frac{1}{\sqrt{1+u^2}}$$ $$\sin x = \frac{2u}{1+u^2}$$ This verifies relation B.

Verify relation C

To verify relation C, we recall the double-angle formula for \(\cos(x)\): $$\cos x = 2\cos^2\left(\frac{x}{2}\right) - 1$$ We already have the expression for \(\cos(x/2)\) from Step 3: $$\cos\left(\frac{x}{2}\right) = \frac{1}{\sqrt{1+u^2}}$$ Now, substitute this expression into the double-angle formula for \(\cos(x)\): $$\cos x = 2\cdot\left(\frac{1}{\sqrt{1+u^2}}\right)^2 - 1$$ $$\cos x = 2\cdot\frac{1}{1+u^2} - 1$$ $$\cos x = \frac{1-u^2}{1+u^2}$$ This verifies relation C.

Key concepts

Rational Integrands

When dealing with integrands that have trigonometric functions in both the numerator and the denominator, a technique called trigonometric substitution can help simplify the expression. By converting these trigonometric integrands into what we call "rational integrands," we make them easier to integrate.One popular substitution involves using the identity \( u = \tan\left(\frac{x}{2}\right) \). This specific substitution transforms the trigonometric terms into rational functions of \( u \). Rational functions are easier to handle in calculus because they only involve polynomials, which are simpler than trigonometric expressions.Here's a brief overview of this transformation mechanism:

• Replace \( x \, \text{with} \, \tan^{-1}(u) \)
• Derive expressions for \( \sin x \) and \( \cos x \) in terms of \( u \)
• These lead to a transformed differential represented as \( dx = \frac{2}{1+u^2} \, du \)

Using this approach, complex trigonometric integrals are converted into simpler rational integrals, making the integration process manageable.

Double-Angle Formulas

The double-angle formulas are crucial identities in trigonometry used to simplify expressions involving trigonometric functions. These are especially important for verifying trigonometric identities or in the solution of integrals as shown in this exercise.The standard double-angle formulas are:

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

In our example, these formulas help convert the half-angle expressions \( \sin\left(\frac{x}{2}\right) \) and \( \cos\left(\frac{x}{2}\right) \) derived from the right triangle into full-angle identities for \( \sin x \) and \( \cos x \).This conversion is vital as it allows one to express \( \sin x \) and \( \cos x \) as rational expressions in terms of \( u \), facilitating the overall simplification of the integration process. Understanding these identities is key in manipulating the angles to derive the desired trigonometric values within the new rational framework set by the substitution.

Right-Triangle Diagrams

Right-triangle diagrams are a visual tool that can simplify the process of verifying trigonometric substitutions and expressions. In trigonometry, these diagrams offer a way to visually understand the relationships between different trigonometric functions based on an angle within a right triangle.In this exercise, we construct a right-angled triangle to verify relations for \( \sin x \) and \( \cos x \) based on the substitution \( u = \tan\left(\frac{x}{2}\right) \). The procedure is as follows:

• Label one of the angles as \( \frac{x}{2} \)
• Assign the opposite side as \( u \) and the adjacent side as 1, which aligns with \( \tan\left(\frac{x}{2}\right) = \frac{\text{opposite}}{\text{adjacent}} \)
• Use the Pythagorean theorem to find the hypotenuse as \( \sqrt{1 + u^2} \)
• With these side lengths, derive \( \sin\left(\frac{x}{2}\right) \) and \( \cos\left(\frac{x}{2}\right) \)

Using these calculations, one can verify the double-angle expressions for \( \sin x \) and \( \cos x \), facilitating their conversion into rational forms. These diagrams are extremely valuable for visualizing and understanding the relationships within trigonometric expressions, providing clarity and simplifying calculations.