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Chapter 8: Problem 62

Exercises \(57-62\) require the use of various trigonometric identities before you evaluate the integrals. $$ \int \sin \theta \sin 2 \theta \sin 3 \theta d \theta $$

Short Answer

Expert verified
The integral evaluates to a combination of cosine terms.

Step by step solution

01

Use Trigonometric Identities

To evaluate the integral, we begin by simplifying the product of sines using trigonometric identities. Specifically, we'll use the product-to-sum identities. For example, using \( \sin A \sin B = \frac{1}{2} [\cos (A-B) - \cos (A+B)] \), we can start simplifying \( \sin \theta \sin 2\theta \).
02

Simplify First Pair Using Product-to-Sum Identity

Rewriting \( \sin \theta \sin 2\theta \), we get: \[\sin \theta \sin 2\theta = \frac{1}{2} \left[ \cos (\theta - 2\theta) - \cos(\theta + 2\theta) \right] = \frac{1}{2} [\cos(-\theta) - \cos(3\theta)]\].Since \(\cos(-\theta) = \cos \theta\), this simplifies to \( \frac{1}{2} \left[ \cos \theta - \cos 3\theta \right] \).
03

Multiply with \(\sin 3\theta\) and Apply Product-to-Sum Again

Now substitute back into the integral: \[ \int \left( \frac{1}{2} [\cos \theta - \cos 3\theta ] \right) \sin 3\theta \, d\theta \].Expand this to: \[\frac{1}{2} \int \left( \cos \theta \sin 3\theta - \cos 3\theta \sin 3\theta \right) d\theta\].Apply the product-to-sum identity for each term separately.
04

Solve Each Integral Separately

Use the product-to-sum identity for each expression:- For \(\cos \theta \sin 3\theta\), use the identity \(\cos A \sin B = \frac{1}{2} [\sin (A+B) - \sin (A-B)]\). - Simplifying, this gives: \[\frac{1}{2} \sin(4\theta) - \frac{1}{2} \sin(2\theta)\].The integral becomes \[\frac{1}{2} \int \frac{1}{2} [\sin(4\theta) - \sin(2\theta)] d\theta\].- For \(\cos 3\theta \sin 3\theta\), it simplifies similarly.
05

Evaluate the Integral

Now, evaluate each term separately using standard integration techniques:- \(\int \sin(4\theta) \, d\theta = -\frac{1}{4} \cos(4\theta)\).- \(\int \sin(2\theta) \, d\theta = -\frac{1}{2} \cos(2\theta)\).Calculate and simplify the full expression for the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are essential tools that simplify expressions involving trigonometric functions. These identities allow us to transform complex trigonometric expressions into simpler forms, making them easier to work with.
Consider the identities like
  • \( an^2 \theta + 1 = \sec^2 \theta \)
  • \( an \theta = \frac{\sin \theta}{\cos \theta} \)
  • \( \sin^2 A + \cos^2 A = 1 \)
These are some basic examples of trigonometric identities that we encounter frequently. In our specific exercise, we deal with three sine functions: \( \sin \theta \), \( \sin 2\theta \), and \( \sin 3\theta \). To evaluate their product, we can often make the expression simpler by converting these into sums or differences of cosines or sines using other identities.
Product-to-Sum Identities
The product-to-sum identities are very useful when dealing with products of trigonometric functions. These identities convert the product of two trigonometric functions into a sum of trigonometric functions. This transformation is crucial when you need to solve integrals.
In particular, the identity for the product of sine functions is:
  • \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \)
Using this identity, we can address the original expression from the exercise: \( \sin \theta \sin 2\theta \). It simplifies using the formula above to \( \frac{1}{2} [\cos(-\theta) - \cos(3\theta)] \), which is further simplified since \( \cos(-\theta) = \cos \theta \). Having a simplified form allows for easier integration as it transforms a complex product into manageable terms.
Integral Calculus
Integral calculus focuses on the process of integration, which is the reverse operation of differentiation. When dealing with integrals involving trigonometric functions, applying identities can simplify the integrals into a more straightforward form.
For example, consider an integral such as \( \int \sin x \, dx \), which simplifies to \( -\cos x + C \), where \( C \) is the constant of integration. In the context of our exercise, beyond simplifying using identities, once the trigonometric functions are rewritten, you directly integrate each term.
The integral of \( \sin 4\theta \) becomes \( -\frac{1}{4}\cos 4\theta \), and for \( \sin 2\theta \), it becomes \( -\frac{1}{2}\cos 2\theta \). These basic integration rules span broader to any trigonometric function as long as they follow their rule of derivatives and antiderivatives.

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Most popular questions from this chapter

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